Lecture 32: Subspaces Related to Linear Transformations (Nicholson Section 5.4)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 23:20
Description: Introduced subspaces that are related to a linear transformation L from R^n to R^m and subspaces that are relevant to an mxn matrix A. Null(L) is a subspace of R^n, Range(L) is a subspace of R^m, Null(A) = “Solution Space of Ax=0” is a subspace of R^n, Col(A) = “span of all columns of A” is a subspace of R^m, Row(A) = “span of all rows of A” is a subspace of R^n.
5:00 --- if [L] is the standard matrix for a linear transformation L, is there any connection between the two subspaces Null(L) and Range(L) and the three subspaces Null([L]), Col([L]), and Row([L])? Answer: Null(L)=Null([L]) and Range(L)=Col([L]). 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 may not 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).
7:15 --- Proved Range(L) is a subspace of R^m.
19:45 --- Did example for L from R^3 to R^3 where L projects a vector x onto the vector [1;2;3]. Found Range(L) and Null(L) using geometric arguments.
