Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 0-32198-238-X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - Problems: 12

Answer

$b$ is not a linear combination of $a_1$, $a_2$, and $a_3$

Work Step by Step

Asking if $b$ is a linear combination of those vectors is equivalent to asking whether the system whose augmented matrix is shown below is consistent. $$ \begin{bmatrix} 1 & 0 & 2 & -5 \\ -2 & 5 & 0 & 11 \\ 2 & 5 & 8 & -7 \end{bmatrix} $$ We can determine this through row reduction. First, add the second row to the third: $$ \begin{bmatrix} 1 & 0 & 2 & -5 \\ -2 & 5 & 0 & 11 \\ 0 & 10 & 8 & 4 \end{bmatrix} $$ Now add twice the first row to the second: $$ \begin{bmatrix} 1 & 0 & 2 & -5 \\ 0 & 5 & 4 & 1 \\ 0 & 10 & 8 & 4 \end{bmatrix} $$ Double the second row and subtract it from the third: $$ \begin{bmatrix} 1 & 0 & 2 & -5 \\ 0 & 5 & 4 & 1 \\ 0 & 0 & 0 & 2 \end{bmatrix} $$ The last row is mathematically impossible ($0=2$), so the matrix is inconsistent and $b$ is not a linear combination of the vectors.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.