Lecture 35: Null(L), Range(L), the rank theorem, Row(A) (Nicholson Section 5.4)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 50:55
Description: Started with a review of the previous lecture: Null(L) and Range(L) for a pair of geometric transformations from R^3 to R^3. For both examples, dim(Null(L)) + dim(Range(L)) = 3. This will be true in general --- if L goes from R^n to R^m we’ll have dim(Null(L)) + dim(Range(L)) = n.
7:55 --- given a linear transformation, L: R^n → R^m, I presented an algorithm for finding a basis for Null(L).
15:45 --- given a linear transformation, L: R^n → R^m, I presented an algorithm for finding a basis for Range(L).
22:40 --- stated the rank theorem dim(Null(L))+dim(Range(L)) = n and explained why it’s true.
25:00 --- defined the “nullity” of a linear transformation/the “nullity” of a matrix.
28:40 --- Defined the row space of A: Row(A). It’s a subspace of R^n.
31:40 --- Stated the theorem that if A and B are row equivalent then Row(A)=Row(B). This means that when you do elementary row operations on a matrix, the row space of the final matrix is the same as the row space of the initial matrix is the same as the row space of each intermediary matrix.
32:45 --- Applied the theorem to a classic exam question.
43:30 --- Discussed a classic mistake that confused/under-rested students make when asked for a basis for Col(A).
45:10 --- Did an example of finding a basis for the solution space S = {x such that Ax=0}.