Linear Transformations of the Plane

How a $2\times 2$ matrix moves every point of $\mathbb{R}^2$

$$ T(\vec v) \;=\; A\vec v, \qquad A\begin{pmatrix}x\\[1pt] y\end{pmatrix} = x\,(A\vec e_1) + y\,(A\vec e_2) $$

The columns of $A$ are the images $A\vec e_1,\;A\vec e_2$ of the standard basis vectors $\vec{e}_{1}$ and $\vec{e}_{2}$.

The plane under $\vec v \mapsto A\vec v$