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?

You need to write $(2,3)$ as a linear combination of the eigenvectors.
In this case, $(2,3) = -(2,1) + 4 (1,1)$, so
$$A \begin{bmatrix}2 \\ 3 \end{bmatrix} = - A  \begin{bmatrix}2 \\ 1 \end{bmatrix} + 4 A  \begin{bmatrix}1\\ 1 \end{bmatrix}$$
and then you can use the fact that these are eigenvectors to easily complete the computation and understand how the "stretching" along the two "eigen-directions" describes the action of $A$ on $(2,3)$.
More generally, you should read about diagonalizing $A$.
