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Unformatted text preview: PRACTICE PROBLEMS Here are some practice problems. Their range of difficulty is considerable. They were thrown together fairly quickly; if you think you see a mistake, let us know. You should not read anything significant into the distribution of questions. For example, there are a lot of diagonalizability questions below. That does not mean that there are a lot diagonalizability questions on the exam. It means that we like syllables. Determine if 1 2 3 4 , 2- 1 2 1 , 1 2 3- 1 , 2- 1 2 4 is linearly independent. A linear transformation T : V W , where V is the set of 3 3 matrices and W = R 4 , has a 6-dimensional null space. Let A be the standard matrix of T . What is the dimension of (Nul A ) ? Let T : R 2 R 2 rotate vectors clockwise by , and then reflect them in the line x + y = 0. Find the standard matrix for T . Let r R n be fixed. Prove that T : R n R n , given by T : x 7 proj r x , is a linear transformation. Is it injective? Surjective?...
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This note was uploaded on 04/06/2010 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.
- Spring '08