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.