University Linear Algebra Change of Basis and Matrix Representations — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 3 definitions and 8 theorems, 5 worked examples, 2 diagrams, in 11 printable pages. This page lists what the notes cover. The notes themselves come with the University Linear Algebra bundle.
Unit notes — in the bundle
Not sold separately. The practice worksheet for this unit is free, with no email and no account.
Change of Basis and Matrix Representations — what the notes cover
6 sections, in the order the unit is taught. Each entry below is the title of a definition or a theorem the notes state in full.
What you need
Also in this section: 1 note on method and common errors.
The change-of-basis matrix
- Definition — Change-of-basis matrix
- Theorem — Converting coordinates
- Theorem — Computing a change-of-basis matrix in n-space
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
The matrix of a linear map relative to two bases
- Definition — Matrix of a linear map relative to two bases
- Theorem — Coordinates of an image
Also in this section: 1 worked example, 1 note on method and common errors.
Matrix representations on polynomial and matrix spaces
- Theorem — Building a matrix representation
- Theorem — Kernel, image and invertibility from a matrix representation
Also in this section: 1 worked example, 1 note on method and common errors.
The change-of-basis formula for an operator
- Theorem — Change-of-basis formula for an operator
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
Similar matrices and similarity invariants
- Definition — Similar matrices
- Theorem — Matrices of one operator are similar
- Theorem — Similarity invariants
Also in this section: 1 worked example, 1 note on method and common errors.
Where the notes are
The notes are one of the documents in the University Linear Algebra Solutions Bundle, beside the unit test, the challenge problems and every worked answer key. The Change of Basis and Matrix Representations practice worksheet is free and stays free.
Every University Linear Algebra topic — the complete Solutions Bundle
One download, one payment, the whole program. For all 9 University Linear Algebra units: the worksheet, the reference notes, the challenge set, the unit test and every answer key — these notes included.
- Worked solutions, not answer lists — every step written out
- Covers the whole year's program at this level
- Less than the price of one hour of tutoring — for the entire year's solutions
Practise it, then test yourself
Notes on their own do not make anything stick. This unit has a free, printable practice worksheet, and an end-of-unit test that comes with the bundle.
Change of Basis and Matrix Representations practice worksheet About the Change of Basis and Matrix Representations unit test