Lecture 17: Representing Linear Transformations as Matrix Transformations (Nicholson Section 2.6/Section 4.4)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 50:39
Description: Started with a discussion of language mapping/transformation/operator.
2:00 --- Revisited shear mappings. Shear in x direction: shear to the right versus shear to the left. Shear in y direction: shear up versus shear down.
13:37 --- Does shearing change area?
16:35 --- The linear transformation T(x) = Proj_[2;1]( x) . Discussed domain, codomain, range, vectors that’re sent to the zero vector by the linear transformation, vectors that’re unchanged by the transformation. All of this was done intuitively; want to do it rigorously.
37:42 --- The transformation that corresponds to reflecting a vector in a given line. Referred students to the book on how to understand this transformation in terms of projections, how to represent it as a matrix transformation and so forth. (Basically, you need to do all the stuff that was done in 16:35-37:41 but for this new transformation.)
39:16 --- The transformation that corresponds to rotating a vector counter-clockwise by a given angle.
47:07 --- Composition of geometric transformation --- how to find a matrix transformation that represents the composition of three geometric transformations. (Note: the same logic would apply for the composition of any number of geometric transformations, not just three. And it’s not limited to geometric transformations; it works for linear transformations in general.)
