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 - 1.8 Exercises: 3

Answer

$\begin{bmatrix}3\\1\\2\end{bmatrix}$ is the unique vector with image $\vec{b}$ under $T$.

Work Step by Step

Finding the vector with a given image under some matrix transformation is equivalent to finding the solution set of a linear system. Hence, we begin with the augmented matrix: $\begin{bmatrix}1&0&-2&-1\\-2&1&6&7\\3&-2&-5&-3\end{bmatrix}$ Add $2$ times row (1) to row (2) and $-3$ times row (1) to row (3): $\begin{bmatrix}1&0&-2&-1\\0&1&2&5\\0&-2&1&0\end{bmatrix}$ Add $2$ times row (2) to row (3): $\begin{bmatrix}1&0&-2&-1\\0&1&2&5\\0&0&5&10\end{bmatrix}$ Divide row (3) by $5$: $\begin{bmatrix}1&0&-2&-1\\0&1&2&5\\0&0&1&2\end{bmatrix}$ Add $-2$ times row (3) to row (2) and $2$ times row (3) to row (1): $\begin{bmatrix}1&0&0&3\\0&1&0&1\\0&0&1&2\end{bmatrix}$ Because there are no free variables, the solution is unique.
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.