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

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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
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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 ....
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