Exact Column/Row Operator Norms, Individual Inputs and Uniform Bounds
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batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-12, -3, -5], [-2, 12, -6]], [14, 15, 11], 15, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-12, -3, -5], [-2, 12, -6]], [14, 15, 11], 15, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
2. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-11, -2, -2], [-1, 11, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-11, -2, -2], [-1, 11, -1]], [12, 13, 3], 13, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-11, -2, -2], [-1, 11, -1]], [12, 13, 3], 13, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
3. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-10, -1, 1], [0, 10, 4]], [10, 11, 5], 11, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-10, -1, 1], [0, 10, 4]], [10, 11, 5], 11, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
4. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-9, 0, 4], [1, 9, -4]], [10, 9, 8], 10, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-9, 0, 4], [1, 9, -4]], [10, 9, 8], 10, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
5. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-8, 1, -4], [2, 8, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-8, 1, -4], [2, 8, 1]], [10, 9, 5], 10, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-8, 1, -4], [2, 8, 1]], [10, 9, 5], 10, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
6. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-7, 2, -1], [-2, 7, 6]], [9, 9, 7], 9, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-7, 2, -1], [-2, 7, 6]], [9, 9, 7], 9, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
7. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-6, 3, 2], [-1, 6, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-6, 3, 2], [-1, 6, -2]], [7, 9, 4], 9, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-6, 3, 2], [-1, 6, -2]], [7, 9, 4], 9, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
8. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-5, -3, 5], [0, 5, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-5, -3, 5], [0, 5, 3]], [5, 8, 8], 8, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-5, -3, 5], [0, 5, 3]], [5, 8, 8], 8, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
9. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-4, -2, -3], [1, 4, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-4, -2, -3], [1, 4, -5]], [5, 6, 8], 8, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-4, -2, -3], [1, 4, -5]], [5, 6, 8], 8, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
10. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-3, -1, 0], [2, 3, 0]], [5, 4, 0], 5, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-3, -1, 0], [2, 3, 0]], [5, 4, 0], 5, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
11. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-2, 0, 3], [-2, 2, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-2, 0, 3], [-2, 2, 5]], [4, 2, 8], 8, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-2, 0, 3], [-2, 2, 5]], [4, 2, 8], 8, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
12. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[-1, 1, -5], [-1, 1, -3]], [2, 2, 8], 8, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-1, 1, -5], [-1, 1, -3]], [2, 2, 8], 8, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
13. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[0, 2, -2], [0, 0, 2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[0, 2, -2], [0, 0, 2]], [0, 2, 4], 4, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 2, -2], [0, 0, 2]], [0, 2, 4], 4, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
14. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[1, 3, 1], [1, -1, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[1, 3, 1], [1, -1, -6]], [2, 4, 7], 7, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 3, 1], [1, -1, -6]], [2, 4, 7], 7, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
15. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[2, -3, 4], [2, -2, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[2, -3, 4], [2, -2, -1]], [4, 5, 5], 5, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -3, 4], [2, -2, -1]], [4, 5, 5], 5, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
16. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[3, -2, -4], [-2, -3, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[3, -2, -4], [-2, -3, 4]], [5, 5, 8], 8, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, -2, -4], [-2, -3, 4]], [5, 5, 8], 8, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
17. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[4, -1, -1], [-1, -4, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[4, -1, -1], [-1, -4, -4]], [5, 5, 5], 5, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, -1, -1], [-1, -4, -4]], [5, 5, 5], 5, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
18. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[5, 0, 2], [0, -5, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[5, 0, 2], [0, -5, 1]], [5, 5, 3], 5, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 0, 2], [0, -5, 1]], [5, 5, 3], 5, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
19. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[6, 1, 5], [1, -6, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[6, 1, 5], [1, -6, 6]], [7, 7, 11], 11, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, 1, 5], [1, -6, 6]], [7, 7, 11], 11, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
20. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[7, 2, -3], [2, -7, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[7, 2, -3], [2, -7, -2]], [9, 9, 5], 9, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 2, -3], [2, -7, -2]], [9, 9, 5], 9, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
21. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[8, 3, 0], [-2, -8, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[8, 3, 0], [-2, -8, 3]], [10, 11, 3], 11, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 3, 0], [-2, -8, 3]], [10, 11, 3], 11, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
22. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[9, -3, 3], [-1, -9, -5]], [10, 12, 8], 12, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, -3, 3], [-1, -9, -5]], [10, 12, 8], 12, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
23. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[10, -2, -5], [0, -10, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[10, -2, -5], [0, -10, 0]], [10, 12, 5], 12, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -2, -5], [0, -10, 0]], [10, 12, 5], 12, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
24. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[11, -1, -2], [1, -11, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[11, -1, -2], [1, -11, 5]], [12, 12, 7], 12, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, -1, -2], [1, -11, 5]], [12, 12, 7], 12, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
25. The Exact l1 Operator Norm Uses Absolute Column Sums
Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.
View answer and explanation
([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum)
Sum absolute values down each column and take the largest sum. The triangle bound is universal, and a coordinate basis vector at a maximizing column attains it. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 0, 1], [2, -12, -3]], [14, 12, 4], 14, maximum_absolute_column_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-l1-norm · candidate
26. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-12, -3, -5], [-2, 12, -6]], [20, 20], 20, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-12, -3, -5], [-2, 12, -6]], [20, 20], 20, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
27. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-11, -2, -2], [-1, 11, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-11, -2, -2], [-1, 11, -1]], [15, 13], 15, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-11, -2, -2], [-1, 11, -1]], [15, 13], 15, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
28. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-10, -1, 1], [0, 10, 4]], [12, 14], 14, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-10, -1, 1], [0, 10, 4]], [12, 14], 14, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
29. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-9, 0, 4], [1, 9, -4]], [13, 14], 14, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-9, 0, 4], [1, 9, -4]], [13, 14], 14, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
30. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-8, 1, -4], [2, 8, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-8, 1, -4], [2, 8, 1]], [13, 11], 13, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-8, 1, -4], [2, 8, 1]], [13, 11], 13, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
31. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-7, 2, -1], [-2, 7, 6]], [10, 15], 15, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-7, 2, -1], [-2, 7, 6]], [10, 15], 15, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
32. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-6, 3, 2], [-1, 6, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-6, 3, 2], [-1, 6, -2]], [11, 9], 11, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-6, 3, 2], [-1, 6, -2]], [11, 9], 11, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
33. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-5, -3, 5], [0, 5, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-5, -3, 5], [0, 5, 3]], [13, 8], 13, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-5, -3, 5], [0, 5, 3]], [13, 8], 13, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
34. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-4, -2, -3], [1, 4, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-4, -2, -3], [1, 4, -5]], [9, 10], 10, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-4, -2, -3], [1, 4, -5]], [9, 10], 10, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
35. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-3, -1, 0], [2, 3, 0]], [4, 5], 5, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-3, -1, 0], [2, 3, 0]], [4, 5], 5, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
36. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-2, 0, 3], [-2, 2, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-2, 0, 3], [-2, 2, 5]], [5, 9], 9, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-2, 0, 3], [-2, 2, 5]], [5, 9], 9, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
37. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[-1, 1, -5], [-1, 1, -3]], [7, 5], 7, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[-1, 1, -5], [-1, 1, -3]], [7, 5], 7, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
38. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[0, 2, -2], [0, 0, 2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[0, 2, -2], [0, 0, 2]], [4, 2], 4, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 2, -2], [0, 0, 2]], [4, 2], 4, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
39. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[1, 3, 1], [1, -1, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[1, 3, 1], [1, -1, -6]], [5, 8], 8, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 3, 1], [1, -1, -6]], [5, 8], 8, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
40. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[2, -3, 4], [2, -2, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[2, -3, 4], [2, -2, -1]], [9, 5], 9, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -3, 4], [2, -2, -1]], [9, 5], 9, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
41. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[3, -2, -4], [-2, -3, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[3, -2, -4], [-2, -3, 4]], [9, 9], 9, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, -2, -4], [-2, -3, 4]], [9, 9], 9, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
42. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[4, -1, -1], [-1, -4, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[4, -1, -1], [-1, -4, -4]], [6, 9], 9, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, -1, -1], [-1, -4, -4]], [6, 9], 9, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
43. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[5, 0, 2], [0, -5, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[5, 0, 2], [0, -5, 1]], [7, 6], 7, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 0, 2], [0, -5, 1]], [7, 6], 7, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
44. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[6, 1, 5], [1, -6, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[6, 1, 5], [1, -6, 6]], [12, 13], 13, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, 1, 5], [1, -6, 6]], [12, 13], 13, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
45. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[7, 2, -3], [2, -7, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[7, 2, -3], [2, -7, -2]], [12, 11], 12, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 2, -3], [2, -7, -2]], [12, 11], 12, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
46. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[8, 3, 0], [-2, -8, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[8, 3, 0], [-2, -8, 3]], [11, 13], 13, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 3, 0], [-2, -8, 3]], [11, 13], 13, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
47. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[9, -3, 3], [-1, -9, -5]], [15, 15], 15, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, -3, 3], [-1, -9, -5]], [15, 15], 15, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
48. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[10, -2, -5], [0, -10, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[10, -2, -5], [0, -10, 0]], [17, 10], 17, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -2, -5], [0, -10, 0]], [17, 10], 17, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
49. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[11, -1, -2], [1, -11, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[11, -1, -2], [1, -11, 5]], [14, 17], 17, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, -1, -2], [1, -11, 5]], [14, 17], 17, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
50. The Exact Maximum-Norm Operator Norm Uses Absolute Row Sums
Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,row_sums,operator_norm,maximum_absolute_row_sum), preserving original row order.
View answer and explanation
([[12, 0, 1], [2, -12, -3]], [13, 17], 17, maximum_absolute_row_sum)
Sum absolute values across each row, then take the maximum. The universal bound is attained by choosing input coordinate signs for a maximizing row; do not interchange row and column formulas. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 0, 1], [2, -12, -3]], [13, 17], 17, maximum_absolute_row_sum).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-linfinity-norm · candidate
51. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-4, -2, -3], proposed C=14. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [-4, -2, -3], [69, 2], 9, 71, 14, 126, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-4, -2, -3], [69, 2], 9, 71, 14, 126, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
52. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-11, -2, -2], [-1, 11, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-3, -1, -2], proposed C=16. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [-3, -1, -2], [39, -6], 3, 39, 16, 48, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-3, -1, -2], [39, -6], 3, 39, 16, 48, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
53. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-2, 0, -1], proposed C=13. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [-2, 0, -1], [19, -4], 3, 23, 13, 39, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-2, 0, -1], [19, -4], 3, 23, 13, 39, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
54. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-1, 1, 0], proposed C=14. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [-1, 1, 0], [9, 8], 1, 9, 14, 14, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-1, 1, 0], [9, 8], 1, 9, 14, 14, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
55. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-8, 1, -4], [2, 8, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[0, 2, 1], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [0, 2, 1], [-2, 17], 3, 19, 9, 27, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [0, 2, 1], [-2, 17], 3, 19, 9, 27, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
56. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[1, -2, 2], proposed C=17. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [1, -2, 2], [-13, -4], 2, 13, 17, 34, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [1, -2, 2], [-13, -4], 2, 13, 17, 34, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
57. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-6, 3, 2], [-1, 6, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[2, -1, 3], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [2, -1, 3], [-9, -14], 6, 23, 9, 54, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [2, -1, 3], [-9, -14], 6, 23, 9, 54, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
58. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-5, -3, 5], [0, 5, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[3, 0, -3], proposed C=14. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [3, 0, -3], [-30, -9], 3, 30, 14, 42, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [3, 0, -3], [-30, -9], 3, 30, 14, 42, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
59. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-4, -2, -3], [1, 4, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[4, 1, -2], proposed C=7. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [4, 1, -2], [-12, 18], 7, 30, 7, 49, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [4, 1, -2], [-12, 18], 7, 30, 7, 49, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
60. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-4, 2, -1], proposed C=5. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [-4, 2, -1], [10, -2], 4, 10, 5, 20, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-4, 2, -1], [10, -2], 4, 10, 5, 20, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
61. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-2, 0, 3], [-2, 2, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-3, -2, 0], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [-3, -2, 0], [6, 2], 5, 8, 9, 45, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-3, -2, 0], [6, 2], 5, 8, 9, 45, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
62. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-2, -1, 1], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [-2, -1, 1], [-4, -2], 2, 4, 9, 18, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-2, -1, 1], [-4, -2], 2, 4, 9, 18, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
63. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[0, 2, -2], [0, 0, 2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-1, 0, 2], proposed C=3. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [-1, 0, 2], [-4, 4], 3, 8, 3, 9, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-1, 0, 2], [-4, 4], 3, 8, 3, 9, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
64. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[1, 3, 1], [1, -1, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[0, 1, 3], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [0, 1, 3], [6, -19], 3, 19, 9, 27, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [0, 1, 3], [6, -19], 3, 19, 9, 27, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
65. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[2, -3, 4], [2, -2, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[1, 2, -3], proposed C=7. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [1, 2, -3], [-16, 1], 6, 17, 7, 42, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [1, 2, -3], [-16, 1], 6, 17, 7, 42, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
66. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[3, -2, -4], [-2, -3, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[2, -2, -2], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [2, -2, -2], [18, -6], 2, 18, 9, 18, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [2, -2, -2], [18, -6], 2, 18, 9, 18, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
67. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[4, -1, -1], [-1, -4, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[3, -1, -1], proposed C=4. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [3, -1, -1], [14, 5], 5, 19, 4, 20, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [3, -1, -1], [14, 5], 5, 19, 4, 20, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
68. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[5, 0, 2], [0, -5, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[4, 0, 0], proposed C=9. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [4, 0, 0], [20, 0], 4, 20, 9, 36, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [4, 0, 0], [20, 0], 4, 20, 9, 36, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
69. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[6, 1, 5], [1, -6, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-4, 1, 1], proposed C=11. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [-4, 1, 1], [-18, -4], 6, 22, 11, 66, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-4, 1, 1], [-18, -4], 6, 22, 11, 66, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
70. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[7, 2, -3], [2, -7, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-3, 2, 2], proposed C=13. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [-3, 2, 2], [-23, -24], 3, 24, 13, 39, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-3, 2, 2], [-23, -24], 3, 24, 13, 39, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
71. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[8, 3, 0], [-2, -8, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[-2, -2, 3], proposed C=10. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [-2, -2, 3], [-22, 29], 7, 51, 10, 70, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [-2, -2, 3], [-22, 29], 7, 51, 10, 70, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
72. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[-1, -1, -3], proposed C=15. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [-1, -1, -3], [-15, 25], 3, 25, 15, 45, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [-1, -1, -3], [-15, 25], 3, 25, 15, 45, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
73. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[10, -2, -5], [0, -10, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[0, 0, -2], proposed C=13. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [0, 0, -2], [10, 0], 2, 10, 13, 26, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [0, 0, -2], [10, 0], 2, 10, 13, 26, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
74. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[11, -1, -2], [1, -11, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, vector=[1, 1, -1], proposed C=19. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(linfinity, [1, 1, -1], [12, -15], 1, 15, 19, 19, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, [1, 1, -1], [12, -15], 1, 15, 19, 19, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
75. One Passing Vector Does Not Certify a Uniform Bound
Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, vector=[2, 2, 0], proposed C=13. Give (norm_name,vector,image,input_norm,output_norm,C,rhs=C*input_norm,pass?,instance_only).
View answer and explanation
(l1, [2, 2, 0], [24, -20], 4, 44, 13, 52, Yes, instance_only)
Compute both norms and compare the one supplied input. The zero vector can pass even with C=0 for a nonzero operator; that does not establish a uniform bound for all vectors. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, [2, 2, 0], [24, -20], 4, 44, 13, 52, Yes, instance_only).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bound-instance · candidate
76. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-12, -3, -5], [-2, 12, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=14. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 15, 14, No, No, bound_too_small)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 15, 14, No, No, bound_too_small).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
77. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-11, -2, -2], [-1, 11, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=16. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 15, 16, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 15, 16, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
78. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 11, 13, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 11, 13, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
79. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-9, 0, 4], [1, 9, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=14. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 14, 14, Yes, Yes, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, 14, Yes, Yes, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
80. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-8, 1, -4], [2, 8, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 10, 9, No, No, bound_too_small)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 10, 9, No, No, bound_too_small).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
81. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-7, 2, -1], [-2, 7, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=17. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 15, 17, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 15, 17, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
82. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-6, 3, 2], [-1, 6, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 9, 9, Yes, Yes, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 9, 9, Yes, Yes, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
83. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-5, -3, 5], [0, 5, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=14. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 13, 14, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 13, 14, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
84. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-4, -2, -3], [1, 4, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=7. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 8, 7, No, No, bound_too_small)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 8, 7, No, No, bound_too_small).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
85. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-3, -1, 0], [2, 3, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=5. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 5, 5, Yes, Yes, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 5, 5, Yes, Yes, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
86. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-2, 0, 3], [-2, 2, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 8, 9, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 8, 9, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
87. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[-1, 1, -5], [-1, 1, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 7, 9, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 7, 9, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
88. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[0, 2, -2], [0, 0, 2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=3. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 4, 3, No, No, bound_too_small)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 4, 3, No, No, bound_too_small).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
89. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[1, 3, 1], [1, -1, -6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 8, 9, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 8, 9, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
90. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[2, -3, 4], [2, -2, -1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=7. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 5, 7, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 5, 7, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
91. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[3, -2, -4], [-2, -3, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 9, 9, Yes, Yes, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 9, 9, Yes, Yes, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
92. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[4, -1, -1], [-1, -4, -4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=4. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 5, 4, No, No, bound_too_small)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 5, 4, No, No, bound_too_small).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
93. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[5, 0, 2], [0, -5, 1]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=9. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 7, 9, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 7, 9, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
94. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[6, 1, 5], [1, -6, 6]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=11. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 11, 11, Yes, Yes, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 11, 11, Yes, Yes, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
95. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[7, 2, -3], [2, -7, -2]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 12, 13, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 12, 13, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
96. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[8, 3, 0], [-2, -8, 3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=10. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 11, 10, No, No, bound_too_small)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 11, 10, No, No, bound_too_small).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
97. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=15. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 15, 15, Yes, Yes, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 15, 15, Yes, Yes, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
98. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[10, -2, -5], [0, -10, 0]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 12, 13, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 12, 13, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
99. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[11, -1, -2], [1, -11, 5]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=linfinity, proposed C=19. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(linfinity, 17, 19, Yes, No, uniform_bound_certificate)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (linfinity, 17, 19, Yes, No, uniform_bound_certificate).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
100. A Usable Uniform Bound May Be Nonoptimal
Use the explicit rectangular integer matrix A=[[12, 0, 1], [2, -12, -3]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. norm_name=l1, proposed C=13. Give (norm_name,exact_operator_norm,C,uniform_bound?,optimal?,boundary). A sufficient C yields uniform_bound_certificate; a smaller C yields bound_too_small.
View answer and explanation
(l1, 14, 13, No, No, bound_too_small)
Compare C with the exact induced norm, not sampled outputs. C at least the norm is a uniform bound; equality, not mere sufficiency, marks optimality. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (l1, 14, 13, No, No, bound_too_small).
Reviewed step-by-step derivation
Bounded linear operator · qf-mod28-operator-bounded-certificate · candidate
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