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Unformatted text preview: M346 First Midterm Exam, September 21, 2004 1. Let V be the subspace of R 4 defined by the equation x 1 + x 2 + x 3 + x 4 = 0. a) Find the dimension of V . b) Find a basis for V . [Any basis will do, but the simpler your answer, the easier part (c) will be. Be sure that each of your vectors really is in V , and that they are linearly independent] c) Let L ( x ) = x 2 x 3 x 4 x 1 . Note that L takes V to V , and can be viewed as an operator on V . Find the matrix [ L ] B , where B is the basis you found in part (b). 2. In R 2 , consider the basis b 1 = 5 3 , b 2 = 3 2 . a) Find the change-of-basis matrices P EB and P BE , where E is the standard basis. b) If v = 13- 2 , find [ v ] B . c) Let L x 1 x 2 = 2 x 2 x 1 + x 2 . Find [ L ] E and [ L ] B . 3. Consider the coupled first-order differential equations dx 1 dt = x 1 +2 x 2 dx 2 dt = 2 x 1 + x 2 Define the new variables y 1 ( t ) = x 1 ( t ) + x 2 ( t ), y 2 ( t ) =...
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