TITLE: Expressing the product Ax as a linear combination of the column vectors of A
QUESTION [10 upvotes]: 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 $\begin{bmatrix}
-13\\\ 
22\\\ 
17
\end{bmatrix}$

REPLY [7 votes]: 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.