Non-square Matrices

...As transformations between dimensions.

Non-Square Matrices
A transformation can reasonably transform inputs into higher dimensions (for example, rotate $90^\circ$ around the $x$-axis) — this would be a nonsquare matrix! Here's an example of one that takes a 2d space and transforms it into a 3d space. Notice that the column space (the span of where the vectors land) has the same number of dimensions as the input space — so the transformation is a full rank transformation. $$A=\begin{bmatrix}2 & 0 \\ -1 & 1 \\ -2 & 1\end{bmatrix}$$ This is a $3\times2$ matrix, and the two columns show that there are 2 basis vectors mapped into three dimensions. A $2\times3$ matrix transforms 3d space into a two-dimensional space, and so it is not full rank.