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!
 A: First: matrices with "two rows, one column" are called vectors (or column vectors). 
Second: What does your function $T$ do to $\mathbf{x}=[1;0]$? What does it do to $\mathbf{x}=[0;1]$?
Or if you prefer:
Third: $A$ will be a $2\times 2$ matrix; write
$$A = \left(\begin{array}{cc}a&b\\c&d\end{array}\right).$$
When you multiply $A$ by $\mathbf{x}$, you get
$$A\mathbf{x} = \left(\begin{array}{cc}
a&b\\
c&d\end{array}\right)\left(\begin{array}{c}x_1\\x_2\end{array}\right) = \left(\begin{array}{cc}
ax_1 + bx_2\\
cx_1 + dx_2
\end{array}\right).$$
In order for this to be the same as
$$T(\mathbf{x}) = 
x_1\left(\begin{array}{r}-3\\5\end{array}\right) + x_2\left(\begin{array}{rr}7\\-2\end{array}\right) = \left(\begin{array}{c}-3x_1 + 7x_2\\
5x_1 -2x_2
\end{array}\right),$$
what are the values of $a$, $b$, $c$, and $d$?
A: 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 $v_1$).
