# Question 17059: Image Membership Uses the Full Real Column Span [Return to catalogue page 86](https://ocme.evemisslab.com/en/questions/page/86/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261102-r03-linear-map-image-foundations/#q-20261014-advance-20261102-r03-linear-map-image-0009) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0033-17059 · q-20261014-advance-20261102-r03-linear-map-image-0009 ## Original question and givens Use this same exact integer matrix A=[[1, 9], [2, 18]] 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. Target=[10, 20]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value]. - G001 — Use this same exact integer matrix A=[[1, 9], [2, 18]] 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. Target=[10, 20]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value]. ## 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](https://ocme.evemisslab.com/en/mko/mko-linear-map-image/) ## Step-by-step derivation S001 · calculationBefore real map determinant diagonal product := mul(1, 18) After real map determinant diagonal product = 18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore real map determinant off-diagonal product := mul(9, 2) After real map determinant off-diagonal product = 18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore real map exact determinant := sub(18, 18) After real map exact determinant = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore same ordered integer matrix=[[1, 9], [2, 18]];usual real map T:R2 to R2;original columns=[[1, 2], [9, 18]] After exact determinant=0;rank=1 Conditions: none A nonzero determinant gives rank2.With determinant0,distinguish the zero matrix from a nonzero singular matrix to get rank0 or1.Keep the same matrix and real vector spaces;do not replace columns by rows or assume the image is the entire codomain. Step identifiers and dependenciesRule: real_map_same_matrix_det_and_nonzero_rank_cases · Uses: G001, S003 S005 · calculationBefore primitive kernel first coordinate := div(9, 1) After primitive kernel first coordinate = 9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore primitive kernel second coordinate := div(-1, 1) After primitive kernel second coordinate = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore primitive kernel canonical sign coordinate0 := mul(1, 9) After primitive kernel canonical sign coordinate0 = 9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore primitive kernel canonical sign coordinate1 := mul(1, -1) After primitive kernel canonical sign coordinate1 = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore other row dependence factor := div(2, 1) After other row dependence factor = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore other row dependence coordinate0 := mul(2, 1) After other row dependence coordinate0 = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore other row dependence coordinate1 := mul(2, 9) After other row dependence coordinate1 = 18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore FIRST nonzero original row index=0;row=[1, 9];gcd=1;raw integer kernel direction=[9, -1];canonical sign=1 After primitive kernel basis=[[9, -1]];other original row=[2, 18]=(2)*selected row Conditions: none Use the first nonzero original row to solve u*x+v*y=0.Divide by the gcd for a primitive integer direction and make its first nonzero coordinate positive.The other row is verified to be a multiple,so both equations reduce to this one rather than a guess from determinant0. Step identifiers and dependenciesRule: real_map_rank1_kernel_canonical_row_and_full_equation · Uses: G001, S011 S013 · calculationBefore canonical kernel basis vector row0 first term := mul(1, 9) After canonical kernel basis vector row0 first term = 9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore canonical kernel basis vector row0 second term := mul(9, -1) After canonical kernel basis vector row0 second term = -9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore canonical kernel basis vector image coordinate0 := sum(9, -9) After canonical kernel basis vector image coordinate0 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore canonical kernel basis vector row1 first term := mul(2, 9) After canonical kernel basis vector row1 first term = 18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore canonical kernel basis vector row1 second term := mul(18, -1) After canonical kernel basis vector row1 second term = -18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore canonical kernel basis vector image coordinate1 := sum(18, -18) After canonical kernel basis vector image coordinate1 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore one nonzero row equation(1)*x+(9)*y=0;nonzero verified kernel vector=[9, -1];chosen nonzero vector coordinate=0 After EVERY real solution is lambda*[9, -1];choose lambda=solution_coordinate0/(9);conversely every real lambda multiple maps to0 Conditions: none If w0 is nonzero,then v is nonzero.Set λ=x/w0;subtract the two zero equations to get v(y−λw1)=0,hence y=λw1.If w0=0,then v=0 and nonzero u forces x=0 while y is arbitrary.Conversely linearity sends every real multiple to zero.This describes the full real kernel,not an integer grid. Step identifiers and dependenciesRule: real_map_rank1_kernel_all_real_solutions_not_one_test · Uses: G001, S018 S020 · calculationBefore other column dependence factor := div(9, 1) After other column dependence factor = 9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore other column dependence coordinate0 := mul(9, 1) After other column dependence coordinate0 = 9 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore other column dependence coordinate1 := mul(9, 2) After other column dependence coordinate1 = 18 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore FIRST nonzero ORIGINAL column index=0;basis vector kept unnormalized=[1, 2];other original column=[9, 18]=(9)*basis After for ANY real(s,t),M(s,t)=((1)*s+(9)*t)*basis;EVERY lambda*basis obtained from input lambda*e0 Conditions: none Every output is a linear combination of the original columns,now verified to lie in this one column's span.Conversely place λ in that original column's input coordinate to obtain λ times the basis.So the image is the full real span.Do not normalize the original column or substitute a row. Step identifiers and dependenciesRule: real_map_rank1_image_both_inclusions_original_column · Uses: G001, S022 S024 · calculationBefore real kernel source dimension minus rank := sub(2, 1) After real kernel source dimension minus rank = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore same real map rank=1;full kernel basis=[[9, -1]];full image basis=[[1, 2]] After nullity=1;image basis cardinality=1;rank plus nullity=2 Conditions: none Count bases of the established full real spaces,not searched grid points.Keep rank,nullity and image-basis count distinct;the same source dimension is2. Step identifiers and dependenciesRule: real_map_full_space_basis_counts_not_grid_counts · Uses: G001, S024 S026 · calculationBefore image rank1 cross first product := mul(1, 20) After image rank1 cross first product = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore image rank1 cross second product := mul(2, 10) After image rank1 cross second product = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore image rank1 column target cross := sub(20, 20) After image rank1 column target cross = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore image rank1 target real coefficient := div(10, 1) After image rank1 target real coefficient = 10 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore image rank1 target representation coordinate0 := mul(10, 1) After image rank1 target representation coordinate0 = 10 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore image rank1 target representation coordinate1 := mul(10, 2) After image rank1 target representation coordinate1 = 20 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore original target=[10, 20];first nonzero original column=[1, 2];cross=0;real coefficient=10 After target=coefficient*basis=[10, 20];real input coefficient*e0=[10, 0] gives target Conditions: none With zero cross product and a nonzero basis,use a nonzero coordinate to solve λ and verify both coordinates.Placing λ in the original column's input slot gives a preimage;λ need not be an integer. Step identifiers and dependenciesRule: image_rank1_full_real_line_membership_not_integer_span · Uses: G001, S031 ## Reverse checks - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Recompute image rank1 target representation coordinate1 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute image rank1 target representation coordinate0 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute image rank1 target real coefficient with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute image rank1 column target cross with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute image rank1 cross second product with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute image rank1 cross first product with exact rational arithmetic, not rounded decimals. - 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 ([10, 20], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: de58b521a79141613831a95b39878e55ba316298acf6fbf36e16a2e4638597f2 - candidate record SHA-256: ea6eb3ae716d3b196fa4dbf8451f671d33e645a75b0b8594e36544c5b5962ede - semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3 - question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261102-r03-linear-map-image-0009.json)