Here's the punchline that ties the whole subject together. Because a linear transformation keeps the grid straight, it is completely determined by where it sends just two vectors: \hat{\imath} = \begin{bmatrix}1\\0\end{bmatrix} and \hat{\jmath} = \begin{bmatrix}0\\1\end{bmatrix}. Tell me where those two land, and I can find where any vector goes — because every vector is just a combination of them.

So we record the transformation by stacking the landing spots of \hat{\imath} and \hat{\jmath} as the columns of a matrix:

That is why a matrix times a vector mixes the columns: the columns are the new homes of the basis vectors.

You choose where the basis lands

The two sliders set where \hat{\imath} (column 1) and \hat{\jmath} (column 2) go. The whole grid follows them rigidly, and the matrix below reads off their coordinates. Place the basis vectors, and you've built the transformation.

Reading a matrix at a glance

From now on you can read a 2×2 matrix as a motion: glance at its first column to see where \hat{\imath} goes, its second to see where \hat{\jmath} goes, and picture the grid stretching to match. Every named transformation that follows — rotation, scaling, shear — is just a particular choice of those two columns.