Visualizing linear algebra

I wanted to learn Dirac’s equation, but linear algebra comes first. Seeing it visually made the concepts click in a way the formulas never did.

Three views of a vector:

1. Physics: an arrow in space, defined only by its length and direction. It can sit anywhere as long as those two stay the same.

2. Computer science: an ordered list of numbers. A house could be the 2D vector (square footage, price). Order matters, and dimension is the length of the list.

3. Mathematics: anything that supports vector addition and scalar multiplication, regardless of how it’s represented.

Linear transformations A transformation is a function that takes a vector in and gives a vector out. It’s linear if grid lines stay parallel and evenly spaced and the origin stays fixed.

Matrices and multiplication A matrix records where the basis vectors land: each column is the landing spot of one basis vector. Matrix multiplication is composing transformations, applied right to left. In AB, B acts first, then A.