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

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