{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# A1 - M ◦ L ≤ rank M b Prove that rank M ◦ L ≤ rank...

This preview shows page 1. Sign up to view the full content.

Math 235 Assignment 1 Due: Wednesday, Sept 22nd 1. Let A = 3 1 4 2 3 5 2 7 3 4 2 - 1 1 3 7 3 2 5 2 1 , then the RREF of A is R = 1 0 1 0 1 0 1 1 0 - 2 0 0 0 1 1 0 0 0 0 0 . a) Find rank( A ) and dim(Null( A )). b) Find a basis for Row( A ). c) Find a basis for Null( A ). d) Find a basis for Col( A ). e) Find a matrix B such that Null( A ) = Col( B ). 2. Let U, V, W be finite dimensional vector spaces over R and let L : V U and M : U W be linear mappings. a) Prove that rank( M L ) rank( M ). b) Prove that rank( M
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: M ◦ L ) ≤ rank( M ). b) Prove that rank( M ◦ L ) ≤ rank( L ). c) Prove that if M is invertible, then rank( M ◦ L ) = rank L . 3. Let T : V → W be a linear mapping and let { ~v 1 ,...,~v r ,~v r +1 ,...,~v n } be a basis for V such that { ~v r +1 ,...,~v n } is a basis for Null( T ). Prove that { T ( ~v 1 ) ,...,T ( ~v r ) } is a basis for the range of T ....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern