...As transformations between dimensions.
Non-Square Matrices
A transformation can reasonably transform inputs into higher
dimensions (for example, rotate
around the
-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.
This is a
matrix, and the two columns show that there are 2 basis vectors
mapped into three dimensions. A
matrix transforms 3d space into a two-dimensional space, and so
it is not full rank.