Linear Algebra 6.3 Matrices for Linear Combinations

This video explains the concept of standard matrices for linear transformations and demonstrates how to find them by examining the transformation’s effect on basis vectors. We’ll also cover combining transformations and representing them using nonstandard bases. This math tutorial aims to clarify how to find the standard matrix and the composition of linear transformations.

Video Chapters:
0:00 Intro
0:10 The Standard Matrix for a Linear Transformation
1:58 Definition of the Standard Matrix for a Linear Transformation
3:01 Finding a Standard Matrix
5:24 Finding a Standard Matrix (on your own)
8:13 Composition of Linear Transformations
10:04 Checking Our Work
11:48 Visualizing
13:27 Composition of Linear Transformations (on your own)
16:55 Visualizing in Reverse
18:07 Inverse of a Linear Transformation
18:54 Finding the Inverse of a Linear Transformation
22:06 Definition of the Transformation Matrix for Nonstandard Bases
23:17 Finding the Image in Two Ways
29:56 Up Next

Textbook: Elementary Linear Algebra 8e, Ron Larson

Slide Decks: https://bellevueuniversity-my.sharepoint.com/:f:/g/personal/kbrehm_bellevue_edu/IgDVnTt2BohsRLsMfseN5FijAXYDd49YlIkHeq5cfmCxG2g?e=ZwTt3c

Playlist: https://youtube.com/playlist?list=PLl-gb0E4MII1RjTV2GegmXFTJwTrek8OI&si=g3Xc3yHB5GkmxMF4