Here's a pet theory i have about linear transformations. Linear transformations are essentially the same as orientation in n-space.

Take 2D transformations, for example. Say you have a hexagon centered on the origin, and you want to scale it so that its height becomes halved while retaining its width. Obviously, this is a straightforward scaling by (1/2,1). But you can also achieve the same effect if you rotate the hexagon into 3-space and project it back to 2-space. To be precise, you rotate it in the YZ plane by 60 degrees, and then project (parallel) the result back to 3-space.

What about shearing? Shearing is essentially rotation in a higher dimension (think about what a rotating square looks like, when viewed from an angle).

What about scaling? Scaling is essentially orientation in 2n dimensions. (Think about how a polygonal face on a 4-polytope's cells can appear reduced in size when the cell is rotated into 4-space.)

What about linear combinations of transformations? They are essentially performing the equivalent orientation operations in distinct additional dimensions. I.e., they are equivalent to orientation in (n+m)-space.

In other words, all linear transformations can be thought of as taking the target object, orienting it appropriately in up to (n+m)-space, and projecting it back to n-space.

What does everyone think about this theory? :-)

If this equivalence is valid, it certainly has very interesting consequences. For example, we would be able to represent linear transformations as a sequence of angles defining the equivalent orientation in (n+m)-space. Conversely, to visualize orientation in (n+m)-space, we can simply reduce it to a linear transformation in n-space.

Does any of this make any sense? :-)