How can a matrix affect one vector but not the other in $\Bbb R^2$ Let $\mathsf T$ be a linear transformation from $\Bbb R^2$ to $\Bbb R^2$ . Let $\vec v_1$, $\vec v_2$, and $\vec w$ be three vectors in $\Bbb R^2$ , as shown below. We are told that $\mathsf T (\vec v_1) = \vec v_1$ and $\mathsf T (\vec v_2) = 3\vec v_2$. On the same axes, sketch $\mathsf T (\vec w)$.
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As a simple example: Consider the matrix $A = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$. 
What's $A\begin{pmatrix} 1 \\ 0 \end{pmatrix}$? What's $A\begin{pmatrix} 0 \\ 1 \end{pmatrix}$?
Now, your matrix is essentially the same, just with a shift of the coordinate system.
