Is b a linear combination of a1, a2, and a3?

Question

Question:

Given $a_1, a_2, a_3,$ and $b$ below, determine if $b$ is a linear combination of $a_1, a_2,$ and $a_3$.

$a_1 = \begin{bmatrix} 1\\ -2\\ 0 \end{bmatrix} $ $a_2 = \begin{bmatrix} 1\\ 0\\ 5 \end{bmatrix} $ $a_3 = \begin{bmatrix} 5\\ -6\\ 8 \end{bmatrix} $

$b = \begin{bmatrix} 2\\ -1\\ 6 \end{bmatrix} $

I create an augmented matrix from the above, and after reducing I get the following matrix.

$\begin{bmatrix} 1 & 0 & 5 & 2\\ 0 & 1 & 4 & 3\\ 0 & 0 & 0 & 0\\ \end{bmatrix} $

Thus I conclude that b is a linear combination of $a_1, a_2,$ and $a_3$. The issue is that the answer to this question in the book is $b$ is not a linear combination of $a_1, a_2$, and $a_3$.

Can someone tell me why this is so? It seems like my answer should be correct.

Answer 1

It is certainly a linear combination as 2 times first column and 3 times second column and 0 times third column equal b.

Probably it was a typo in your book.