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(a) Write one of the four vectors -3 3 2 8 as a linear combination of some or all of the others. (b) Find a linearly independent subset of {ū,ū, w,7} that has the same span (in other words, find a basis for the subspace spanned by these vectors).

(a) Write one of the four vectors -3 3 2 8 as a linear combination of some or all of the others. (b) Find a linearly independent subset of {ū,ū, w,7} that has the same span (in other words, find a basis for the subspace spanned by these vectors).

Advanced Engineering Mathematics
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 provide a solution using the answer key I have attached below. 

For part (a), if you form the coefficient matrix and row-reduce it, you will find that its 0 –1 3 RREF is 0 0 0 1 2 -2 It follows from examining this RREF that one possible answer is w = -ū+20, although there are several other correct answers. For part (b), since the first two columns of u i w have pivots in the RREF, a basis for span{ū, v, w, 7} is {ū, v}. (Again, other correct answers are possible: as long as your set consists of exactly two of the vectors ū, ī, ū, ã, it is a basis for their span, since no two of these vectors are scalar multiples of each other and therefore all six possible pairs of them are linearly independent sets.)
Transcribed Image Text:For part (a), if you form the coefficient matrix and row-reduce it, you will find that its 0 –1 3 RREF is 0 0 0 1 2 -2 It follows from examining this RREF that one possible answer is w = -ū+20, although there are several other correct answers. For part (b), since the first two columns of u i w have pivots in the RREF, a basis for span{ū, v, w, 7} is {ū, v}. (Again, other correct answers are possible: as long as your set consists of exactly two of the vectors ū, ī, ū, ã, it is a basis for their span, since no two of these vectors are scalar multiples of each other and therefore all six possible pairs of them are linearly independent sets.)
(a) Write one of the four vectors w = 8. as a linear combination of some or all of the others. (b) Find a linearly independent subset of {u, ū, w,i} that has the same span (in other words, find a basis for the subspace spanned by these vectors).
Transcribed Image Text:(a) Write one of the four vectors w = 8. as a linear combination of some or all of the others. (b) Find a linearly independent subset of {u, ū, w,i} that has the same span (in other words, find a basis for the subspace spanned by these vectors).
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