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

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 40

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online