Finding the matrix of linear transformation $$T\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}= \begin{pmatrix} 3x_1+5x_2\\ 7x_1-x_2\end{pmatrix}$$ $$T:{\mathbb R}^2 \rightarrow {\mathbb R}^2$$
I have no idea how to start this. Any help would be appreciated. I need to find the transformation matrix T.
One way to do this is to make an ansatz (educated guess) of the form of $T$.  You can assume that 

$$
T \;\; =\;\; \left [ \begin{array}{cc}
a &amp; b \\
c &amp; d \\
\end{array} \right ]
$$

and now compute that 

$$
T \left [ \begin{array}{c}
x_1 \\
x_2\\
\end{array} \right ] \;\; =\;\; \left [ \begin{array}{cc}
a &amp; b \\
c &amp; d \\
\end{array} \right ] \left [ \begin{array}{c}
x_1 \\
x_2\\
\end{array} \right ] \;\; =\;\; \left [ \begin{array}{c}
ax_1 + bx_2 \\
cx_1 + dx_2 \\
\end{array} \right ]
$$

by ordinary matrix multiplication.  What now must be the values of $a,b,c,d$?
The matrix is: $A_{T} = \begin{pmatrix} 3 &amp; 5 \\ 7 &amp; -1 \end{pmatrix}$. The matrix has the columns $T(e_1),T(e_2)$. And what are $T(e_1) = T((1,0)), T(e_2) = T((0,1))$ ?