2009_Winter_Practice_Midterm_2

2009_Winter_Practice_Midterm_2 - Math 20F Practice Midterm...

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Math 20F Practice Midterm 2. Problem 1. True or False: For each statement below, determine whether it is true or false, and circle the appropriate letter. You do not need to justify your answer. ( T F ) If A is a (4 × 6) matrix and dim(Nul( A ))= 2, then A x = b has a solution for every b R 4 . ( T F ) The set { ± x y ² | xy 0 } is a subspace of R 2 . ( T F ) If A is a square matrix and det A = 0, then 0 is an eigenvalue for A T A . ( T F ) Let v 1 , v 2 , v 3 , v 4 be vectors in a vector space V , and let H = Span { v 1 , v 2 , v 3 , v 4 } . If dim H = 3, and v 1 - v 3 = v 4 , then { v 1 , v 2 , v 3 } is a basis for H .

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Problem 2. Let A = ± - 2 12 - 1 5 ² . a) Find the eigenvalues of A . b) Find a diagonal matrix D and an invertible matrix P such that A = PDP - 1 .
Problem 3. a) Find all values for k such that det A = 0, where A = 1 1 - 1 2 3 k 1 k 3 b) Find all values for k such that the following system of equations has more than one solution: x 1

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2009_Winter_Practice_Midterm_2 - Math 20F Practice Midterm...

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