Lecture 21: Usefulness of the determinant: invertibility and geometry (Nicholson Section 3.2/Section 4.4)
Sub Project: Linear Algebra Lecture Videos
Video Link: Lecture 21: Usefulness of the determinant: invertibility and geometry (Nicholson Section 3.2/Section 4.4)
Alternate Video Access via MyMedia | Video Duration: 49:15
Description: Review of how the three elementary row operations affect the determinant.
6:50 --- If A is equivalent to B by some sequence of elementary row operations then det(B) equals some nonzero number times det(A). It follows that A is invertible if and only if det(A) is nonzero.
16:30 --- How determinants interact with products of square matrices: det(AB) = det(A)det(B). It follows that det(AB) = det(BA) and that if A is not invertible then AB and BA aren’t invertible either.
19:00 --- How to remember that det(AB) = det(A)det(B).
21:00 --- Did a classic midterm question involving determinants.
27:00 ---How to use determinant to compute the area of a parallelogram. Discussed why the absolute value is needed.
35:00 --- How the area of the image of a region under a linear mapping is determined by the determinant of the standard matrix for the linear mapping and the area of the region. The previous book introduced “standard matrix” early on which is why I’m referring to in these lectures; Nicholson only introduces it in chapter 9. So you don’t know this language. Here’s what “standard matrix” means. Nicholson refers to “the matrix of a linear transformation” at the bottom of page 106. This is the “standard matrix”; he just doesn’t call it that until page 497 (he’s trying to avoid confusing you too early, I assume). I proved this for a parallelogram and stated it for general regions in the plane. Note: the proof for general regions in the plane is a multivariable Calculus thing, not a linear algebra thing.
41:00 --- Example where the linear mapping is rotation.
43:30 --- Example where the linear mapping is reflection about a line.
47:00 --- Example where the linear mapping is projection onto a line.
