Lect19

# Lect19 - Chapter 3. Vector Spaces Math1111 Row Space &...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems Theorem Let A , B be be matrices. If B = MA for some matrix M , then r ( B ) ⊂ r ( A ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems Theorem Let A , B be be matrices. If B = MA for some matrix M , then r ( B ) ⊂ r ( A ) . Proof. If B = MA , each row of B is a linear combination of rows of A . ∴ Each row of B belongs to span of rows of A , i.e. belongs to r ( A ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems Theorem Let A , B be be matrices. If B = MA for some matrix M , then r ( B ) ⊂ r ( A ) . Proof. If B = MA , each row of B is a linear combination of rows of A . ∴ Each row of B belongs to span of rows of A , i.e. belongs to r ( A ) . ∵ r ( A ) is a subspace, ∴ Linear combinations of rows of B belong to r ( A ) . Hence, r ( B ) ⊂ r ( A ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems Theorem Let A , B be be matrices. If B = MA for some matrix M , then r ( B ) ⊂ r ( A ) . Theorem 3.6.1 If A is row equivalent to B , then r ( A ) = r ( B ) . Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems Theorem Let A , B be be matrices. If B = MA for some matrix M , then r ( B ) ⊂ r ( A ) . Theorem 3.6.1 If A is row equivalent to B , then r ( A ) = r ( B ) . Application To study r ( A ) , we can instead study r ( U ) where U is a row echelon form of A . Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems Theorem Let A , B be be matrices. If B = MA for some matrix M , then r ( B ) ⊂ r ( A ) . Theorem 3.6.1 If A is row equivalent to B , then r ( A ) = r ( B ) . Application To study r ( A ) , we can instead study r ( U ) where U is a row echelon form of A . Example . Find a basis for r ( A ) if A =       1 1 2 2 1 2 3 4 2 3 4 5       Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems (Cont’d) Question Is c ( A ) = c ( B ) if A is row equivalent to B ? Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems (Cont’d) Question Is c ( A ) = c ( B ) if A is row equivalent to B ? Example . Let A =       1- 2 1 1 2- 1 3 2- 2 1 1 3 4 1 2 5 13 5       and U =       1- 2 1 1 2 1 1 3 1       . Given that U is a row echelon form of A , and let a i = the i th column of A . (i) Show that c ( A ) 6 = c ( U ) . (ii) Show that a 1 , a 2 , a 5 are linearly independent. (iii) Show that a 1 , a 2 , a 3 are linearly dependent. Chapter 3. Vector Spaces Math1111 Row Space & Column space Theorems (Cont’d) Question Is c ( A ) = c ( B ) if A is row equivalent to B ? Ans. No Example . Let A =       1- 2 1 1 2- 1 3 2- 2 1 1 3 4 1 2 5 13 5       and U =       1- 2 1 1 2 1 1 3 1       ....
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## This note was uploaded on 04/30/2010 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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Lect19 - Chapter 3. Vector Spaces Math1111 Row Space &...

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