eigenvector transformation Say you have the following matrix A in $R^2 \rightarrow R^2$:
$
\begin{bmatrix}
7 & -10 \\
5 & -8
\end{bmatrix}
$
Thus the eigenvalues/eigenvectors are: 2 $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ and -3 $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$.
Thus the eigenspace matrix is $\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$. 
Say you have the vector v(x,y) of (2,3), thus Ax = [-16, -14].   
I'm confused as to how does the eigenspace and eigenvalues allow me to easily see what A is doing to the vector (2,3)? 
How do I apply the eigenvalues/eigenspace on vector v(2,3) to see what A is doing to it?

Keep in mind that $A(2,1) = 2(2,1)$ and $A(1,1) = -3(1,1)$ by def of eigenvalues and eigenvectors.  Can you write the desired vector $(2,3)$ as a linear combination of $(2,1)$ and $(1,1)$?  Note that $-1(2,1)+4(1,1) = (2,3)$.
Hence, you can simplify the calculation to
$$A(2,3) = -1*2*(2,1)+4*(-3)*(1,1) = (-4,-2)-(12,12) = (-16,-14).$$
