Concept guide
The parent ring U is the ring of 2-by-2 upper-triangular matrices over 𝔽₂ (elements written (a,b,c) for the matrix with first row (a,b) and second row (0,c)), already qualified as a finite unital ring. (1) Prove that I={(0,0,0),(0,1,0)} is a two-sided ideal of U. (2) Prove that L={(0,0,0),(1,0,0)} satisfies left absorption but is not a two-sided ideal. (3) Prove that the diagonal subring D={(a,0,c)} contains 1 but is not an ideal. (4) Show that {0} and U itself are two-sided ideals.
Finite two-sided ideal
derivation-mko-finite-two-sided-ideal-01
Concept guide
On V=ℝ² take the ℓ¹ norm ‖x‖₁=|x₁|+|x₂| and the metric induced by that same norm, d₁(x,y)=‖x−y‖₁. (1) Prove that ‖·‖₁ is a norm and d₁ a metric, and compute d₁((3,−1),(1,2)). (2) Prove completely, from the Cauchy completeness of the reals, that (ℝ²,d₁) is complete, so (ℝ²,‖·‖₁) is a real Banach space; also cite the named theorem that finite-dimensional real normed spaces are complete. (3) Prove that ‖·‖₁ violates the parallelogram law and hence is induced by no inner product.
Real Banach space
derivation-mko-real-banach-space-01
Concept guide
Let M be the symmetric integer matrix with first row (2,1) and second row (1,3), and ⟨x,y⟩_M=xᵀMy for x,y∈ℝ². (1) Prove separately that ⟨·,·⟩_M is bilinear, symmetric and positive definite, hence an inner product on ℝ²; positive definiteness is certified universally by the 2-by-2 Sylvester conditions, with both sufficiency and necessity proved in full. (2) Decide why the forms given by N (rows (1,2),(2,1)), P (rows (1,1),(1,1)) and R (rows (2,1),(0,3)) are not inner products.
Real inner-product space
derivation-mko-real-inner-product-space-01
Concept guide
Let A be the integer matrix with first row (1,−2) and second row (3,4), and T(x)=Ax:ℝ²→ℝ². (1) With the ℓ^∞ norm at both ends, prove that ‖Ax‖_∞≤7‖x‖_∞ for all x and that no smaller constant works, so ‖T‖_{∞→∞}=7 (the maximum absolute row sum). (2) With the ℓ¹ norm at both ends, prove ‖T‖_{1→1}=6 (the maximum absolute column sum). (3) Explain three traps: non-optimal constants, finitely many inputs, and unbounded range.
Bounded linear operator
derivation-mko-bounded-linear-operator-01
Concept guide
(1) Joint table J₁: X∈{0,1}, Y∈{0,1,2}, P(0,0)=1/12, P(0,1)=1/12, P(0,2)=1/6, P(1,0)=1/6, P(1,1)=1/6, P(1,2)=1/3. Aggregate both marginals from the joint table, check the product equation at all 6 value pairs, and decide whether X and Y are independent; also prove that independence implies independence of any coordinate events {X∈S} and {Y∈T} and E[XY]=E[X]E[Y]. (2) Joint table J₂: X uniform on {−1,0,1} and Y=X². Prove Cov(X,Y)=0 although X and Y are not independent. (3) Use joint table J₃ to show that checking a single value pair is not enough.
Independence of random variables
derivation-mko-random-variable-independence-01
Concept guide
Let G=(V,E) with V={0,1,2,3,4,5,6} and E consisting of exactly the five edges {0,1}, {0,2}, {1,2}, {3,4}, {4,5}. First prove that 'joined by a path' is an equivalence relation on V; then compute each equivalence class by complete BFS, prove that the set BFS returns is exactly that class, and conclude Q(G)={{0,1,2},{3,4,5},{6}}, where the isolated vertex 6 forms a singleton component.
Connected component
derivation-mko-connected-component-01
Concept guide
Using complete multiplication tables, decide that ℤ/5ℤ is a finite integral domain and ℤ/6ℤ is not; then prove, as a theorem that comes after the definition, that every nonzero element of any finite integral domain is invertible, using finiteness; find all inverses in ℤ/5ℤ and explain why 2 has no inverse in ℤ/6ℤ.
Finite integral domain
derivation-mko-finite-integral-domain-01
Concept guide
Over R=ℤ/4ℤ take f=1+2x and g=2x. Using the coefficient-sequence definition, compute and normalize f+g, f·g, f·f and g·g; prove the top-coefficient formula (fg)_{m+n}=f_m g_n and explain why deg(fg)=deg f+deg g fails here; finally prove that R[x] is not a finite ring.
Polynomial ring over a finite commutative ring
derivation-mko-polynomial-ring-over-finite-commutative-ring-01
Concept guide
In any real inner-product space, prove that N(v)=√⟨v,v⟩ is a norm: nonnegativity and separation, absolute homogeneity, and the triangle inequality (via a fully proved Cauchy–Schwarz inequality). Then, on ℝ² with the inner product given by A=[[2,1],[1,2]], compute exactly the squared norms and the squared distance for x=(1,0) and y=(0,1), and check Cauchy–Schwarz, the triangle inequality and the parallelogram identity on this instance.
Inner-product-induced norm
derivation-mko-inner-product-induced-norm-01
Concept guide
On Ω={0,1,2,3,4} with masses 1/8, 3/8, 1/4, 1/4, 0 and X taking the values 4, 0, 2, 6, 10 in order, let G be generated by the partition {0,1}, {2,3}, {4}. From the two defining conditions of conditional expectation (G-measurable, same integral on every event of G), find all versions of E[X|G]: prove they must equal 1 and 4 on the two positive-mass blocks and may take any value on the zero-mass block {4} (determined only almost surely), and explain why pointwise conditioning, checking only the total expectation, and the unweighted mean are all wrong.
Finite conditional expectation
derivation-mko-conditional-expectation-01
Concept guide
On Ω={0,1,2,3} (two coin tosses: 0=HH, 1=HT, 2=TH, 3=TT), represent the σ-algebras at times 0, 1, 2 by partition rows P_0, P_1, P_2. Prove that 'the later partition refines the earlier one ⇔ the σ-algebras are nested', check every adjacent pair of times completely and conclude that (F_t) is a filtration; give a sequence of partitions that is not a filtration; and show that a filtration neither makes every process adapted nor implies independence.
Finite filtration
derivation-mko-finite-filtration-01
Concept guide
Let p=1/3, and let the cylinder certificate give each finite binary word w (length n, with k ones) the probability p^k(1−p)^{n−k}. Prove that this family is consistent with total mass 1 at every level; obtain from it (with the cited Kolmogorov existence theorem) a probability on {0,1}^ℕ; show that every X_i has the complete marginal Bernoulli(1/3) and that every finite subfamily is independent. Then use three contrasts to show that a checked prefix, pairwise independence and equal expectations are each insufficient for an infinite IID claim.
Independent identically distributed sequence
derivation-mko-iid-sequence-01
Concept guide
On X={0,1,2,3,4,5}, R is given by 12 explicitly listed ordered pairs. First certify completely that R is an equivalence relation, then compute every class from the definition [a]={x∈X ∣ aRx}; prove that aRb ⇒ [a]=[b], that any two classes are equal or disjoint and that the classes cover X, giving K(R)={{0,3},{1,4},{2,5}}. Finally use S=R∪{(0,1),(1,0)} to show that a relation failing transitivity cannot form equivalence classes.
Equivalence class
derivation-mko-equivalence-class-01
Concept guide
F3 is a field and 2/2=1; nonzero multiplication maps supply inverses in a finite domain.
Finite field
derivation-mko-finite-field-01
Concept guide
Verify the consistent inner product, norm, metric and completeness of R^2 with A=[[2,1],[1,2]].
Real Hilbert space
derivation-mko-real-hilbert-space-01
Concept guide
Audit every martingale obligation on one space.
Finite martingale
derivation-mko-martingale-01
Concept guide
X/R={{0,2},{1,3}}, and projection values are whole classes.
Quotient set
derivation-mko-quotient-set-01
Concept guide
There is an injection A to B with |A|=2<3=|B|; a reverse surjection is not injective.
Cardinality comparison of sets
derivation-mko-cardinality-comparison-01
Concept guide
Compute the CDF on the whole real line, including its right-continuous values and its jumps.
Cumulative distribution function
derivation-mko-cumulative-distribution-function-01
Concept guide
Prove convergence in probability on a common finite space.
Convergence in probability
derivation-mko-convergence-in-probability-01
Concept guide
N0 times {0,1,2} injects into N0 by c(n,j)=3n+j and is at most countable.
At-most-countable set
derivation-mko-at-most-countable-set-01
Concept guide
Audit every bipartition obligation and distinguish one failed partition from nonbipartiteness.
Bipartite graph
derivation-mko-bipartite-graph-01
Concept guide
Prove distributional convergence at continuity points of the limiting CDF.
Convergence in distribution
derivation-mko-convergence-in-distribution-01
Concept guide
I={0,2} is a proper two-sided ideal; R/I has two cosets and well-defined operation tables.
Quotient ring of a finite ring
derivation-mko-finite-quotient-ring-01