Advanced Linear Algebra - Lecture 10: The Standard Matrix of a Linear Transformation

We learn how to represent linear transformations as matrices.

Please leave a comment below if you have any questions, comments, or corrections.

[secondaryschoolscience]

I'm a reader of your book "Introduction to Linear and Matrix Algebra" published by Springer. I love your book, which is amongst the best-in-class for someone like me who like to have more in-depth knowledge on linear algebra beyond those studied in high school. In particular, I love the section 1.4.2 "A Catalog of Linear Transformation". However, during my re-visit on this topic, I try to derive some of the formulae by myself, and am successful doing so except rotation in 3D. I'm able to get the exact formulae for Ryz and Rxy, but not Rzx which I have the signs for those "sine's" swapped. I observe you've put some remark at the margin to use Rzx instead of Rxz, but not sure if this is correct. My approach is: if Rzx means to have y being the axis of rotation, and positive z rotates towards positive x, then Rzx(e3) would become (sin(theta), 0, cos(theta) and Rzx(e1) become (cos(theta), 0, -sin(theta)). More or less, by analogy to the 2D case (by ignoring y), z replacing x and x replacing y. As a result, I have the signs of those sine's different from yours. Would you please shed me some light on where my approach goes wrong. Thanks a lot!

[dougwalker]

Excellent video. The use of standard matrices with coordinate vectors and transformations is just the best explanation I have seen for this.