Expressing the product Ax as a linear combination of the column vectors of
$A$=
$\begin{bmatrix}
4 & 0 & -1\\
3 & 6 & 2\\
0 & -1 & 4
\end{bmatrix}$
$\vec{x}$=$\begin{bmatrix}
-2\\
3\\
5
\end{bmatrix}$
I get it now. They just want me to multiply the two vectors together.
I end up with $\begin{bmatrix}
-13\\\
22\\\
17
\end{bmatrix}$
Best Answer
I have edited your question to make sure it is understood correctly. So if I understand correctly, you have a matrix $A$ = $\begin{bmatrix} 4 & 0 & -1\\ 3 & 6 & 2\\ 0 & -1 & 4 \end{bmatrix}$ and a vector $\vec{x}$ = $\begin{bmatrix} -2\\ 3\\ 5 \end{bmatrix}$. You are trying to write the product $A$$\vec{x}$ as a linear combination of the column vectors of $A$. Now to do that, you need to perform the multiplication by its very definition: $\begin{bmatrix} 4 & 0 & -1\\ 3 & 6 & 2\\ 0 & -1 & 4 \end{bmatrix}$ $\cdot$ $\begin{bmatrix} -2\\ 3\\ 5 \end{bmatrix}$ actually means $-2$$\begin{bmatrix} 4\\ 3\\ 0 \end{bmatrix}$ $+$ $3$$\begin{bmatrix} 0\\ 6\\ -1 \end{bmatrix}$ $+$ $5$$\begin{bmatrix} -1\\ 2\\ 4 \end{bmatrix}$, which is what I believe your question is asking for.