Matrices as transformations

At school matrices show up as tables with strange multiplication rules. This is a different view: a matrix is a movement of space. Rotating, stretching, shearing. Everything you can do to a grid without tearing or bending it.

Two arrows are enough

The plane has two basic vectors: $\hat{\imath} = (1, 0)$ points right and $\hat{\jmath} = (0, 1)$ points up. Every other vector is a combination of them, for example $(3, 2)$ means “three times $\hat{\imath}$ and twice $\hat{\jmath}$”.

That is exactly what you see in the playground below. The red arrow is $\hat{\imath}$, the green one is $\hat{\jmath}$. Drag their tips.

Matrices are columns

We write the coordinates of the new arrows side by side and a matrix appears. The first column is where $\hat{\imath}$ landed, the second is where $\hat{\jmath}$ landed.

import numpy as np

A = np.array([[0, -1],
[1,  0]])   # rotation by 90°

v = np.array([3, 2])
print("A @ v =", A @ v)

# determinant: how many times the area changes
print("det A =", np.linalg.det(A))