Overview
On this page, we discuss changing the basis for our choice of coordinates, and the special case of diagonalizing a matrix, which is possible when there is a basis for \(\mathbb{R}^n\) consisting of \(n\) linearly independent eigenvectors of this matrix.
Important
The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class - but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Use a basis change matrix to convert coordinate vectors given with respect to one basis to another.
- Use a basis change matrix to convert the matrix representation of a linear transformation with respect to one choice of input/output bases to another choice.
- Find all eigenvalues and eigenvectors of a matrix (review).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Verify that a set of eigenvectors form a basis (“eigenbasis“) for the full space.
- Use a basis change matrix with an eigenbasis to diagonalize a matrix, if possible.
To prepare for class
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Watch this video by 3Blue1Brown which explains the fundamentals about applying a basis change, i.e. re-expressing coordinates with respect to a new basis:
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Watch the last part (starting at 13:00) of this video by 3Blue1Brown which explains how we can diagonalize a matrix when its eigenvectors form a basis (“eigenbasis”) for the full space, and why this is useful:
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Watch this video by Trefor Bazett which goes through a full example of diagonalizing a \(3\times 3\) matrix:
