section7 - 10/7/11 EE221A Section 7 1 Practice midterm...

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EE221A Section 7 10/7/11 1 Practice midterm Problem 1. Injectivity and surjectivity a) Suppose that T : V W is an injective, linear map, and that { v 1 ,...,v n } is a linearly independent set in V . Prove that { T ( v 1 ) ,T ( v 2 ) ,...,T ( v n ) } is a linearly independent set in W . b) True or false: If L : X Y is a surjective, linear map, and { x 1 ,...,x n } spans X , then { L ( x 1 ) ,...,L ( x n ) } spans Y . Please support your answer by a proof or coun- terexample. Problem 2. Subspaces. True or false: If X 1 and X 2 are subspaces of a vector space X , then X 1 X 2 is also a subspace of V . Please support your answer by a proof or counterexample. Problem 3. Matrix Representation of Linear Maps (4 points). Consider the vector space R 2 , and the linear map R θ which takes any vector in R 2 and rotates it in a counterclockwise direction about the origin through an angle θ . (a) Derive the matrix representation of this linear map with respect to the standard
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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