change of basis
(2.3 hours to learn)
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.
This concept has the prerequisites:
- 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)
→ Khan Academy: Linear Algebra
Location: Lecture sequence "Change of basis"
→ 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)
→ MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
- create concept: shift + click on graph
- change concept title: shift + click on existing concept
- link together concepts: shift + click drag from one concept to another
- remove concept from graph: click on concept then press delete/backspace
- add associated content to concept: click the small circle that appears on the node when hovering over it
- other actions: use the icons in the upper right corner to optimize the graph placement, preview the graph, or download a json representation