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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).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.6: Rank Of A Matrix And Systems Of Linear Equations
Problem 69E: Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of...
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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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