Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.8 Exercises - Page 69: 11

Answer

Yes, this $\vec{b}$ is in the range of this linear transformation.

Work Step by Step

We simply row reduce the augmented matrix $[\mathbf{A}\ \ \vec{b}]$ and find that there are no rows of the form $[0\ 0\ 0\ 0\ b]$ with $b\neq 0$. Hence, the equation $T(\vec{x})=\mathbf{A}\vec{x}=\vec{b}$ is consistent. You should find that the equation has infinitely many solutions of the form $\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{bmatrix}=\begin{bmatrix}3+9s-7t\\1+4s-3t\\s\\t\end{bmatrix}$
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