Question 17198: Nullity Counts a Full Kernel Basis

Original question and givens

Use this same exact integer matrix A=[[1, 23], [2, 46]] 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 (rank,nullity,kernel_basis).

• G001 — Use this same exact integer matrix A=[[1, 23], [2, 46]] 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 (rank,nullity,kernel_basis).

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-kernel

Step-by-step derivation

real map determinant diagonal product := mul(1, 46)

real map determinant diagonal product = 46

real map determinant off-diagonal product := mul(23, 2)

real map determinant off-diagonal product = 46

real map exact determinant := sub(46, 46)

real map exact determinant = 0

same ordered integer matrix=[[1, 23], [2, 46]];usual real map T:R2 to R2;original columns=[[1, 2], [23, 46]]

exact determinant=0;rank=1

primitive kernel first coordinate := div(23, 1)

primitive kernel first coordinate = 23

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

primitive kernel second coordinate = -1

primitive kernel canonical sign coordinate0 := mul(1, 23)

primitive kernel canonical sign coordinate0 = 23

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, 23)

other row dependence coordinate1 = 46

FIRST nonzero original row index=0;row=[1, 23];gcd=1;raw integer kernel direction=[23, -1];canonical sign=1

primitive kernel basis=[[23, -1]];other original row=[2, 46]=(2)*selected row

canonical kernel basis vector row0 first term := mul(1, 23)

canonical kernel basis vector row0 first term = 23

canonical kernel basis vector row0 second term := mul(23, -1)

canonical kernel basis vector row0 second term = -23

canonical kernel basis vector image coordinate0 := sum(23, -23)

canonical kernel basis vector image coordinate0 = 0

canonical kernel basis vector row1 first term := mul(2, 23)

canonical kernel basis vector row1 first term = 46

canonical kernel basis vector row1 second term := mul(46, -1)

canonical kernel basis vector row1 second term = -46

canonical kernel basis vector image coordinate1 := sum(46, -46)

canonical kernel basis vector image coordinate1 = 0

one nonzero row equation(1)*x+(23)*y=0;nonzero verified kernel vector=[23, -1];chosen nonzero vector coordinate=0

EVERY real solution is lambda*[23, -1];choose lambda=solution_coordinate0/(23);conversely every real lambda multiple maps to0

other column dependence factor := div(23, 1)

other column dependence factor = 23

other column dependence coordinate0 := mul(23, 1)

other column dependence coordinate0 = 23

other column dependence coordinate1 := mul(23, 2)

other column dependence coordinate1 = 46

FIRST nonzero ORIGINAL column index=0;basis vector kept unnormalized=[1, 2];other original column=[23, 46]=(23)*basis

for ANY real(s,t),M(s,t)=((1)*s+(23)*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=[[23, -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, 1, [[23, -1]])

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.