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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please explain part a,b and c using the answer key provided.
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).
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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