Matrices as Transformations

Stop thinking of a matrix as a grid of numbers; think of it as a function that moves vectors: $\mathbf{x} \mapsto A\mathbf{x}$. The columns of $A$ tell you everything: they are the images of the basis vectors. If $A\mathbf{e}_1 = \text{col}_1$ and $A\mathbf{e}_2 = \text{col}_2$, linearity determines where every other vector goes.

To rotate the plane 90° counterclockwise: $\mathbf{e}_1 = (1,0) \mapsto (0,1)$ and $\mathbf{e}_2 = (0,1) \mapsto (-1,0)$, so