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Consider the vectors 10 11 -3 -22 1 -20 9. -32 (a) These vectors are linearly dependent. Write one of them as a linear combination of the others. (b) Find a linearly independent subset of this set of vectors that has the same span (in other words, find a basis for the subspace spanned by these vectors).

Consider the vectors 10 11 -3 -22 1 -20 9. -32 (a) These vectors are linearly dependent. Write one of them as a linear combination of the others. (b) Find a linearly independent subset of this set of vectors that has the same span (in other words, find a basis for the subspace spanned by these vectors).

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Please explain part a and b. ( I have attached an answer key).

Consider the vectors 10 11 8 -20 32 (a) These vectors are linearly dependent. Write one of them as a linear combination of the others. (b) Find a linearly independent subset of this set of vectors that has the same span (in other words, find a basis for the subspace spanned by these vectors).
Transcribed Image Text:Consider the vectors 10 11 8 -20 32 (a) These vectors are linearly dependent. Write one of them as a linear combination of the others. (b) Find a linearly independent subset of this set of vectors that has the same span (in other words, find a basis for the subspace spanned by these vectors).
1 -2 10 11 -3 4 8 , you will find that its -22 For part (a), if you row-reduce the coefficient matrix 1 -20 2 -32 1 0 2 1 -4 0 It follows from examining this RREF that one possible answer is 1 RREF is 0 10 1 -22 -3 - 4 4 = 2 -32 although other correct answers are possible. For part (b), since the first, second, and fourth columns of the coefficient matrix have pivots in the RREF, a basis for the span of all four columns consists of these three only: 11 8 9.
Transcribed Image Text:1 -2 10 11 -3 4 8 , you will find that its -22 For part (a), if you row-reduce the coefficient matrix 1 -20 2 -32 1 0 2 1 -4 0 It follows from examining this RREF that one possible answer is 1 RREF is 0 10 1 -22 -3 - 4 4 = 2 -32 although other correct answers are possible. For part (b), since the first, second, and fourth columns of the coefficient matrix have pivots in the RREF, a basis for the span of all four columns consists of these three only: 11 8 9.
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