W13Ex2 - Math 217 Midterm 2 Winter 2013 Solutions Name...

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Math 217 – Midterm 2 Winter 2013 Solutions Name: Section: Question Points Score 1 12 2 6 3 7 4 5 5 10 6 9 7 15 8 8 9 8 Total: 80
Math 217 Solutions Midterm 2, Page 1 of 9 1. Write complete, precise definitions for each of the following (italicized) terms. (a) (3 points) The n × n matrix A is diagonalizable .
(b) (3 points) An eigenvector of a square matrix A .
(c) (3 points) A basis for a subspace V of R n .
(d) (3 points) Let g : X Y be a function between sets X and Y . The function g is one-to-one .
Math 217 Solutions Midterm 2, Page 2 of 9 2. Let T be the function from Mat 2 × 2 ( R ) to Mat 2 × 2 ( R ) given by T ( D ) = [ 1 1 2 2 ] D. (a) (3 points) Find a basis for Range( T ). A correct answer with no justification will receive full credit. (b) (3 points) Find a basis for ker( T ). A correct answer with no justification will receive full credit.
Math 217 Solutions Midterm 2, Page 3 of 9 3. (7 points) Let V be a vector space with bases B = { b 1 , b 2 , b 3 } and C = { c 1 , c 2 , c 3 } , and let T : V V be a linear transformation. Suppose that b 1 = c 1 b 2 = 2 c 1 + c 2 b 3 = c 2 + c 3 and [ T ] C = 1 0 1 0 2 0 0 0 1 . Compute [ T ] B .
4. (5 points) Let V

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