Prac-Final - Practice Final Math 235 Spring 2009 1 Consider...

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Practice Final, Math 235 Spring 2009 1. Consider the matrix A = 1 0 2 - 1 2 0 1 1 3 - 2 1 - 3 . Let b = b 1 b 2 b 3 b 4 . Find equations in b 1 ,b 2 ,b 3 ,b 4 so that the equation Ax = b can be solved. Find a basis of the image of A . 2. True or false? Justify. If W is a subspace of an n -dimensional vector space V and dim ( W ) = n , the W = V . There exists a linear transformation T : R 5 R 3 whose kernel has dimension 1. Let R 2 × 2 be the vector space of 2 × 2-matrices. The function det : R 2 × 2 R , which maps a matrix A to its determinant det ( A ), is linear. Let P be the vector space of polynomials in x , and let W = ± p ( x ) : xp ( x ) - 2 Z 1 0 p ( t ) t = 0 ² . Then W is a subspace of P . 3. Let T : R 2 R 2 with T ³´³ 1 0 µ¶µ = " 2 2 2 2 !# and T ³´³ 0 1 µ¶µ = " - 2 2 2 2 !# . (a) Calculate T ³´³ 1 1 µ¶µ . (b) Find the matrix
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Prac-Final - Practice Final Math 235 Spring 2009 1 Consider...

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