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: 13

Answer

$b$ is not a linear combination of the column vectors of $A$.

Work Step by Step

The question posed is equivalent to asking whether the system represented by the augmented matrix below is consistent: $$ \begin{bmatrix} 1 & -4 & 2 & 3 \\ 0 & 3 & 5 & -7 \\ -2 & 8 & -4 & -3 \\ \end{bmatrix} $$ We can do this via row reduction. Begin by adding twice the first row to the third row: $$ \begin{bmatrix} 1 & -4 & 2 & 3 \\ 0 & 3 & 5 & -7 \\ 0 & 0 & 0 & 3 \\ \end{bmatrix} $$ We see the contraction $0=3$; thus, the system is inconsistent and $b$ is not a linear combination of the columns of $A$.
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