Question 17014: The Image Basis Comes from Original Matrix Columns

Original question and givens

Use this same exact integer matrix A=[[1, 14], [2, 28]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,image_basis,full_real_column_span).

• G001 — Use this same exact integer matrix A=[[1, 14], [2, 28]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,image_basis,full_real_column_span).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-linear-map-image

Step-by-step derivation

real map determinant diagonal product := mul(1, 28)

real map determinant diagonal product = 28

real map determinant off-diagonal product := mul(14, 2)

real map determinant off-diagonal product = 28

real map exact determinant := sub(28, 28)

real map exact determinant = 0

same ordered integer matrix=[[1, 14], [2, 28]];usual real map T:R2 to R2;original columns=[[1, 2], [14, 28]]

exact determinant=0;rank=1

primitive kernel first coordinate := div(14, 1)

primitive kernel first coordinate = 14

primitive kernel second coordinate := div(-1, 1)

primitive kernel second coordinate = -1

primitive kernel canonical sign coordinate0 := mul(1, 14)

primitive kernel canonical sign coordinate0 = 14

primitive kernel canonical sign coordinate1 := mul(1, -1)

primitive kernel canonical sign coordinate1 = -1

other row dependence factor := div(2, 1)

other row dependence factor = 2

other row dependence coordinate0 := mul(2, 1)

other row dependence coordinate0 = 2

other row dependence coordinate1 := mul(2, 14)

other row dependence coordinate1 = 28

FIRST nonzero original row index=0;row=[1, 14];gcd=1;raw integer kernel direction=[14, -1];canonical sign=1

primitive kernel basis=[[14, -1]];other original row=[2, 28]=(2)*selected row

canonical kernel basis vector row0 first term := mul(1, 14)

canonical kernel basis vector row0 first term = 14

canonical kernel basis vector row0 second term := mul(14, -1)

canonical kernel basis vector row0 second term = -14

canonical kernel basis vector image coordinate0 := sum(14, -14)

canonical kernel basis vector image coordinate0 = 0

canonical kernel basis vector row1 first term := mul(2, 14)

canonical kernel basis vector row1 first term = 28

canonical kernel basis vector row1 second term := mul(28, -1)

canonical kernel basis vector row1 second term = -28

canonical kernel basis vector image coordinate1 := sum(28, -28)

canonical kernel basis vector image coordinate1 = 0

one nonzero row equation(1)*x+(14)*y=0;nonzero verified kernel vector=[14, -1];chosen nonzero vector coordinate=0

EVERY real solution is lambda*[14, -1];choose lambda=solution_coordinate0/(14);conversely every real lambda multiple maps to0

other column dependence factor := div(14, 1)

other column dependence factor = 14

other column dependence coordinate0 := mul(14, 1)

other column dependence coordinate0 = 14

other column dependence coordinate1 := mul(14, 2)

other column dependence coordinate1 = 28

FIRST nonzero ORIGINAL column index=0;basis vector kept unnormalized=[1, 2];other original column=[14, 28]=(14)*basis

for ANY real(s,t),M(s,t)=((1)*s+(14)*t)*basis;EVERY lambda*basis obtained from input lambda*e0

real kernel source dimension minus rank := sub(2, 1)

real kernel source dimension minus rank = 1

same real map rank=1;full kernel basis=[[14, -1]];full image basis=[[1, 2]]

nullity=1;image basis cardinality=1;rank plus nullity=2

Reverse checks

• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute real kernel source dimension minus rank with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute other column dependence coordinate1 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute other column dependence coordinate0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute other column dependence factor with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute canonical kernel basis vector image coordinate1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute canonical kernel basis vector row1 second term with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute canonical kernel basis vector row1 first term with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute canonical kernel basis vector image coordinate0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute canonical kernel basis vector row0 second term with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute canonical kernel basis vector row0 first term with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute other row dependence coordinate1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute other row dependence coordinate0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute other row dependence factor with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute primitive kernel canonical sign coordinate1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute primitive kernel canonical sign coordinate0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute primitive kernel second coordinate with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute primitive kernel first coordinate with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute real map exact determinant with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute real map determinant off-diagonal product with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute real map determinant diagonal product with exact rational arithmetic, not rounded decimals.

Result

([[1, 14], [2, 28]], 1, [[1, 2]], full_real_column_span)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.