Linear Algebra: Find a matrix A such that T(x) is Ax for each x I am having difficulty solve this problem in my homework:
(In my notation, $[x;y]$ represents a matrix of 2 rows, 1 column)

Let $\mathbf{x}=[x_1;x_2]$, $v_1$=[−3;5] and $v_2=[7;−2]$ and let $T\colon\mathbb{R}^2\to\mathbb{R}^2$  be a linear transformation that maps $\mathbf{x}$ into $x_1v_1+x_2v_2$. Find a matrix $A$ such that $T(\mathbf{x})$ is $A\mathbf{x}$ for each $\mathbf{x}$.

I am pretty clueless. So I assume that I start off with the following:
$x_1v_1 + x_2v_2 = x_1[−3;5] + x_2[7;−2]$
But I do not know what to do from here, or if this is even the correct start!

If I understand you correctly, I would say that 
$$A = \left(\begin{array}{rr}-3&7\\5&-2\end{array}\right) \ \textrm{and} \ x'=Ax.$$
You can see this if you use 
$$x' = \left(\begin{array}{cc}x_1\\x_2\end{array}\right).$$
Then $$x_1'= -3\cdot x_1 + 7\cdot x_2 = x_1 \cdot v_{11} + x_2\cdot v_{21}$$ and $$x_2'= 5\cdot x_1-2\cdot x_2 = x_1\cdot v_{12} + x_2\cdot v_{22}$$ (here $v_{12}$ is the second element of the first $v_1$).
