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# Final - Exam I Math 235 Name Instructor 12 0 1 a What does...

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Exam I Math 235 March 11, 2009 Name: Instructor 1. a: What does it mean for a vector v to be in the image of the matrix A = 1 2 0 0 1 1 1 0 - 2 ? b: Find a linear equation in v 1 , v 2 and v 3 so that the system Aw = v 1 v 2 v 3 has a solution. 2. Let the transformation T : R 2 R 2 be the rotation of the plane through π/ 2. a: Find T 1 0 and T 0 1 . In other words, find the matrix of T . b: Find the smallest positive integer k such that T k is the identity transformation. 3. True or false? You must justify your answer to receive credit. a: Let P 3 be the vector space of polynomials in x of degree less than 4. The subset S = { p ( x ) : p ( x ) + xp ( x ) = 0 } is a subspace of P 3 . b: If T : R 4 R 3 is a linear transformation and the vectors 1 1 0 , 2 3 5 are a basis for im ( T ), then dim ( ker ( T )) = 2. 4. Let V be the vector space spanned by the independent functions cos x and sin x . Let G : V V be the linear transformation defined as G ( f ( x )) = f ( x ) - f ( x ) for f ( x ) V.

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Final - Exam I Math 235 Name Instructor 12 0 1 a What does...

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