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Consider the 6 × 8 matrix 3 0 0 0 0 1 2 1 0 0 0 0 1 0 0 -3 4 -7 -9 A = 0 0 0 0 0 0 0 0 0 1 8 1 and let ā1,..., ās E Rº be the columns of A (in left-to-right order). (a) Does ā7 belong to the span of đ1, ā2, ā3, đ4, āz, and ā6? If not, explain why not; if so, write đz explicitly as a linear combination of these six vectors. (b) Exactly one of the following three sets of vectors is linearly dependent: {ã1, ā3, đ4, ā6}, {ã1,ã3,āz, ā7}, {ã1, ã3, đ4, ā5}. Identify which one it is, and find an explicit linear dependence relation among the vectors in it. (c) Determine dim col(A) and dim null(A).

Consider the 6 × 8 matrix 3 0 0 0 0 1 2 1 0 0 0 0 1 0 0 -3 4 -7 -9 A = 0 0 0 0 0 0 0 0 0 1 8 1 and let ā1,..., ās E Rº be the columns of A (in left-to-right order). (a) Does ā7 belong to the span of đ1, ā2, ā3, đ4, āz, and ā6? If not, explain why not; if so, write đz explicitly as a linear combination of these six vectors. (b) Exactly one of the following three sets of vectors is linearly dependent: {ã1, ā3, đ4, ā6}, {ã1,ã3,āz, ā7}, {ã1, ã3, đ4, ā5}. Identify which one it is, and find an explicit linear dependence relation among the vectors in it. (c) Determine dim col(A) and dim null(A).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.8: Determinants
Problem 34E
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Please explain part a,b and c using the answer key provided.

Consider the 6 × 8 matrix [1 3 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1 -3 1 4 -7 -9 A = 1 1 0 0 and let ā1,..., ās E Rº be the columns of A (in left-to-right order). (a) Does ā7 belong to the span of đ1, ā2, ā3, đ4, āz, and ā6? If not, explain why not; if so, write đz explicitly as a linear combination of these six vectors. (b) Exactly one of the following three sets of vectors is linearly dependent: {ã1, ã3, đ4, ā6}, {ã1, ā3, ā5, ā7}, {ã1, đ3, đ4, ã5}. Identify which one it is, and find an explicit linear dependence relation among the vectors in it. (c) Determine dim col(A) and dim null(A).
Transcribed Image Text:Consider the 6 × 8 matrix [1 3 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1 -3 1 4 -7 -9 A = 1 1 0 0 and let ā1,..., ās E Rº be the columns of A (in left-to-right order). (a) Does ā7 belong to the span of đ1, ā2, ā3, đ4, āz, and ā6? If not, explain why not; if so, write đz explicitly as a linear combination of these six vectors. (b) Exactly one of the following three sets of vectors is linearly dependent: {ã1, ã3, đ4, ā6}, {ã1, ā3, ā5, ā7}, {ã1, đ3, đ4, ã5}. Identify which one it is, and find an explicit linear dependence relation among the vectors in it. (c) Determine dim col(A) and dim null(A).
For part (a), the answer is yes, ā7 does belong to the span of ā1, d2, ā3, đ4, đz, and ã6. An explicit representation is d7 = 5ã1 + 0ã2 + 6ã3 - 7đ4 + Ođ5 + 8ās. | For part (b), the linearly dependent set is {ā1, ā3, ã4, āz}, and a nontrivial linear dependence relation among these vectors is 2ā1 – 3ā3 + 4ā4 – āz = 0. For part (c), dim col(A) = 5 (the number of pivot columns) and dim null(A) = 3 (the number of non-pivot columns).
Transcribed Image Text:For part (a), the answer is yes, ā7 does belong to the span of ā1, d2, ā3, đ4, đz, and ã6. An explicit representation is d7 = 5ã1 + 0ã2 + 6ã3 - 7đ4 + Ođ5 + 8ās. | For part (b), the linearly dependent set is {ā1, ā3, ã4, āz}, and a nontrivial linear dependence relation among these vectors is 2ā1 – 3ā3 + 4ā4 – āz = 0. For part (c), dim col(A) = 5 (the number of pivot columns) and dim null(A) = 3 (the number of non-pivot columns).
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