> Open question I still have - what is the "geometric interpretation" of the transpose operation? A^T? AFAIK, there is not a solid geometric interpretation. Pa

> Open question I still have - what is the "geometric interpretation" of the transpose operation? A^T? AFAIK, there is not a solid geometric interpretation. Part of the trouble is that the transpose can change the shape of the matrix. For example, for a vector the transpose turns it into a matrix outputting a single number. These two objects seem to be rather incomparable geometrically. The best intuition I have for transposition is that it represents a time reversal (but not necessarily an inverse). In the case of a vector, you have to think of it as a linear transformation that maps 1 to that vector. The transpose instead maps that vector to its length squared. I have more vague intuition for the why its length squared and how it relates to projections, but its hard to put into words. With rotation matrices, this time reversal results in the inverse. So essentially rotation/skewing is reversed, but scaling is not.