section44 - Chapter 4: The Vector Space R n MAT188H1F Lec03...

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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, its 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 didnt 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 row-echelon 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 ....
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section44 - Chapter 4: The Vector Space R n MAT188H1F Lec03...

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