Context: I have a question about affine transform matrices in 3-space. Matrices are 4x4, with the right-most column being the translation, and the bottom row being [0,0,0,1].
In discussions I read about applying affine transform matrices, it's pointed out that if you want to transform a point, make the last field (W) 1 (so that the translation is applied), while if you want to transform a vector, make the last field 0 (so that the translation is not applied).
Something I never see discussed is the fact that this rule about W in vectors applies to columns of affine transform matrices. The first column is the new X axis. That is, <1,0,0> will be transformed to be whatever is the first column of the matrix. The remaining columns are the new Y axis, the new Z axis, and the new origin. The new axes don't have to be orthogonal, so you can get any desired manner of skewing. Also notice how the bottom entries of each of those columns are 0 for vectors and 1 for the point.
Now, I'm sure that real math people here already know all this and can probably give some insight into why nobody ever mentions it. Is it just totally obvious? If so, why are so many other obvious things pointed out but not this?
Anyhow, here's my main question: Is there any interesting interpretation of the rows?
When computing an inner product, you can think of the result as the cosine of the angle between the two vectors or the magnitude of the projection of one vector onto the other. So does it mean something to project a vector onto the X components of another set of vectors? Anyhow, this is about as far as I can go with this, so I'd love it someone who gets this better could go deeper into the meanings of the rows.
Thanks!