This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 4: The Vector Space R n MAT188H1F Lec03 Burbulla Chapter 4 Lecture Notes Fall 2007 Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n Chapter 4: The Vector Space R n 4.4: Rank Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.4: Rank Three Subspaces Determined by a Matrix Let A by an m × n matrix. Suppose its columns are C 1 , C 2 , . . . , C n and its rows are R 1 , R 2 , . . . , R m . Define three subspaces: 1. row A = span { R 1 , R 2 , . . . , R m } , which is a subspace of R n , called the row space of A . 2. col A = span { C 1 , C 2 , . . . , C n } , which is a subspace of R m , called the column space of A . 3. null A = { X ∈ R n  AX = O } , which is a subspace of R n , called the null space of A . Of course, we have mentioned the null space of A before. And the column space of A is just the image of A : col A = span { C 1 , C 2 , . . . , C n } = im A . Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.4: Rank The Row Space of A That means really only the row space of A is new. Or is it? Actually, it’s the subspace of the three above that we have used most often! When you use elementary row operations to reduce a matrix, you are taking multiples of one row and adding them to another. That means you are working with linear combinations of the rows of A . Indeed, if a sequence of row operations is used to reduce the matrix A to R , then row A = row R . What you are actually doing when you row reduce a matrix is finding a basis for the row space of A . We just didn’t look at it that way when we started this course. Aside: similarly, if you use column operations to reduce a matrix A , then you are working in the column space of A . Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.4: Rank Example 1 Let A = 1 3 1 2 1 1 1 2 1 1 10 1 1 ; row A is a subspace of R 5 . Reduce A (as in Chapter 1) to its reduced rowechelon form, R : 1 3 1 2 1 1 1 2 1 1 10 1 1  2 R 1 + R 2→ R 1 + R 3 1 0 3 1 0 1 7 1 0 0 1 7 1 0 → R 2 + R 3 1 0 3 1 0 1 7 1 0 0 0 The rows of R are just linear combinations of the rows of A . So row A = row R ....
View
Full Document
 Fall '07
 Burbula
 Linear Algebra, vector space Rn

Click to edit the document details