## Linear Algebra and Its Applications (5th Edition)

Published by Pearson

# 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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