---
title: "Change of Basis"
description: "The same map has different matrices in different bases. A change-of-basis matrix converts coordinates, and similar matrices are the same map seen from two viewpoints."
canonical: https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484
source: https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484.md
retrieved: 2026-09-12
---

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# Change of Basis

The same map has different matrices in different bases. A change-of-basis matrix converts coordinates, and similar matrices are the same map seen from two viewpoints.

Subject: Mathematics · Area: Linear Algebra · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484

## Ready when they can

- Convert coordinates between two bases with a transition matrix
- Write the matrix of a map in a new basis by conjugating
- Explain what stays the same about a map when the basis changes

## Lesson: Same map, new coordinates

The same vector has different coordinates in different bases, and the transition matrix is the dictionary between the languages. Its columns are the old basis vectors written in the new coordinates. Example: (3, 1) = 2(1, 1) + 1(1, minus 1), so its B-coordinates are (2, 1). Label every vector with its basis.

**Example.** The same map has different matrices too: A new = P inverse A P, read as translate, apply, translate back. With A = [[1, 2], [0, 3]] and P = [[1, 1], [0, 1]], the new matrix is [[1, 0], [0, 3]], whose (1, 2) entry is 0. Conversion runs through an inverse, so it works in both directions.

The map stays fixed while only its description changes, so some quantities are immune. Trace, determinant, and eigenvalues belong to the transformation itself. A diag(2, 5) map has trace 7 in every basis, and no basis change moves its determinant.

**Tip.** Most change of basis errors come from untracked labels. Write the basis next to every coordinate vector before you multiply. If a result looks strange, check first whether two languages got mixed in one equation.

**Recap.** Convert coordinates with P, move matrices with P inverse A P, and trust trace, determinant, and eigenvalues to survive the trip.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [The Inverse of a Matrix and When It Exists](https://lightmysky.com/learn/mathematics/the-inverse-of-a-matrix-and-when-it-exists-mt_1K6lcfMQ_b)
- [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto)

## Opens up

- [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa)