JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Matrices as Transformations

Stop seeing a matrix as a grid of numbers and start seeing it as a verb: a function that moves every point of space at once, while keeping grid lines straight and the origin fixed. The secret is that the columns simply record where the basis vectors land.

A matrix is a verb, not a noun

Read a matrix as a linear transformation: a function that takes in a vector and moves it to a new place. Apply it to every point at once and you watch all of space shift, rotate, stretch, or squash. The word *linear* promises two things: grid lines stay straight and evenly spaced, and the origin never moves.

The columns are where the basis vectors land

Here is the key trick. The transformation matrix is built column by column: the first column is where (1,0) goes, the second column is where (0,1) goes. To transform any vector, scale those two landing spots by the vector's coordinates and add — that is exactly matrix-times-vector.

A = [[2,1],[0,3]]
(1,0) lands at (2,0)  <- first column
(0,1) lands at (1,3)  <- second column
so A*(x,y) = x*(2,0) + y*(1,3) = (2x+y, 3y)
Read the columns as landing spots, and matrix-times-vector becomes obvious.

A small zoo of transformations

Once you read columns as landing spots, common transformations become recognizable on sight. Notice how each one just describes where (1,0) and (0,1) end up.

  1. Scaling [[2,0],[0,3]]: stretch x by 2, y by 3 — a clean zoom that keeps the axes as axes.
  2. Rotation by 90 degrees [[0,-1],[1,0]]: (1,0) swings to (0,1) and (0,1) swings to (-1,0) — the whole plane turns.
  3. Shear [[1,1],[0,1]]: (1,0) stays put but (0,1) slides to (1,1) — squares tilt into parallelograms.