# change of basis

(2.3 hours to learn)

## Summary

Sometimes it's convenient to perform computations in a basis other than the standard one. Change of basis matrices can be used to convert vectors and matrices from one basis to another.

## Context

This concept has the prerequisites:

- bases
- linear transformations as matrices (Change of basis is often used for finding better representations of linear transformations.)
- matrix multiplication (Change of basis is represented in terms of a matrix product.)
- matrix inverse (The change of basis formula includes a matrix inverse.)
- vector spaces (Some canonical examples of change of basis are for vector spaces other than R^n.)

## Goals

- Understand what it means to represent a linear transformation in a basis other than the standard one

- Be able to convert matrices and vectors between bases algebraically

## Core resources (read/watch one of the following)

## -Free-

→ Khan Academy: Linear Algebra

## -Paid-

→ Introduction to Linear Algebra

An introductory linear algebra textbook with an emphasis on numerical methods.

Location:
Section 7.2, "The matrix of a linear transformation," pages 384-393, and Section 7.3, "Diagonalization and the pseudoinverse," subsection "Similar matrices," pages 399-401

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 9.1, "Linear transformations and change of basis," pages 413-420

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ MIT Open Courseware: Linear Algebra (2011)

Videos for an introductory linear algebra course focusing on numerical methods.

## See also

- Change of basis transformations are important for defining equivalence classes on matrices: The spectral decomposition of a symmetric matrix can be viewed as a change of basis.