# Linear transformation

A function on vectors that keeps addition and scaling intact. Every such function in coordinates is multiplication by a matrix.

Maths (university) · Part 09: Determinants, inverses and vector spaces · Card 9 of 9 · Last updated 2026-10-19

## Analogy

A funhouse mirror that stretches and turns every point but keeps straight lines straight.

## Example

**Rotate by 90 degrees**

Matrix [[0,-1],[1,0]].
(1,0) goes to (0,1).
(0,1) goes to (-1,0).
(2,3) goes to (-3,2).

## In practice

Rotations, reflections and scalings in graphics and robotics are all linear transformations.

## Questions people ask

### How do I find the matrix of a linear transformation?

See where the transformation sends each basis vector, and use those results as the columns. A 90 degree rotation sends (1,0) to (0,1) and (0,1) to (-1,0), so the matrix columns are (0,1) and (-1,0). Linearity then handles every other vector automatically, since each is a combination of the basis vectors.

### Why are translations not linear transformations?

A linear transformation must send the zero vector to itself. Moving every point by adding (1,2) shifts the origin, so it breaks the rule. Graphics software handles translations with a trick using one extra coordinate, so it can still use matrices. Rotation, scaling and reflection do keep the origin fixed, so they are linear.

## Related concepts

- [Rank and dimension](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-8)
- [Inverse matrix](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-3)
- [Solving with the inverse](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-4)
- [Vector space](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-5)

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