[zawerf]

I use the 2nd point a lot for debugging 3d transforms. To expand upon it, for example in three dimensions the three axes are:

(1, 0, 0)

(0, 1, 0)

(0, 0, 1)

To find out where those axes are after a 3x3 matrix transform, you just read off the first, second, and third columns of the matrix respectively. Then you can mentally visualize another unit cube in the new coordinate system using those three vectors as the edges of the cube.

Really basic change-of-basis stuff but academic lectures don't emphasize how useful it is to be able to look at a matrix of numbers immediately know what it does.

[aidos]

This concept totally changed my intuitive understanding of matrices. Beautifully illustrated in the below 3blue1brown video.

https://youtu.be/kYB8IZa5AuE?t=3m15s