184: Matrix Transform
Title text: In fact, draw all your rotational matrices sideways. Your professors will love it! And then they'll go home and shrink.
A rotational matrix transformation (i.e. the big brackets with "cos" and "sin" in them) is used in computer graphics to rotate an image. In general, to rotate a point [a1, a2] in a 2D space by z° clockwise, you can multiply it by the rotation matrix [[cos z°, sin z°], [-sin z°, cos z°]]. In this case, the left side of the equation is rotating [a1, a2] by 90°. Simplifying the trigonometry, the 90° clockwise rotation matrix is [[0, 1], [-1, 0]], so multiplying this by [a1, a2], you should get [a2, -a1].
The joke is that the author performed the rotation transformation on the notation rather than on the vector it represents.
Rotational matrix transformations are a special case of the general linear matrix transform, which can do other things to images, including the other two affine transformations of scaling them or translating (moving) them. On a pedantic note, normally mathematics uses counterclockwise as a default, although computer graphics frequently use a clockwise default, so this may be an intentional reference.
So the title text may be referring to the professors going home (translation) and shrinking (scaling) from the joke; it may also refer to them going home and seeing a shrink (counselor) in despair of their student understanding.
The red text in xkcd: volume 0 notes that mean people wrote into state that the direction of rotation was wrong, but that's due to readers not checking the signs. Rather, the actual typo was that "shrink" in the title-text should have been "drink", but was left because of (1/2)[a1, a2].
- [A square matrix next to a vertical two-by-one matrix, equated to a horizontal matrix that looks like the two-by-one matrix turned 90 degrees.]
- [Square matrix:
- cos90° sin90°
- -sin90° cos90°]
- [Two by one matrix:]
- [An equal sign]
- [The same two by one matrix, but rotated by 90 degrees clockwise:]
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