Spring 2005 Final

# Spring 2005 Final - Math 221 Final Exam Name Section No...

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Math 221 - Final Exam- May 17, 2005 Name: Section: No notes. No calculators. No books. WORK + ANSWER = CREDIT 1. (15) Suppose A is a 4 × 4 matrix whose inverse is A - 1 = - 6 9 - 5 1 9 - 1 - 5 2 - 5 - 5 9 - 3 1 2 - 3 1 . Find x such that Ax = 1 1 1 1 . 1

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2. (20) Answer the following questions. Give your reasoning. a) Is the transformation T a b c d = a b c d + 2 3 1 5 from R 2 × 2 to R 2 × 2 linear? (Here R 2 × 2 denotes the space of all 2 × 2- matrices.) b) Is the transformation T ( f ( t )) = f ( t ) + 2 f ( t ) from P 2 to P 2 linear? (Here, P 2 denotes the space of all polynomials f ( t ) of degree 2.) c) Is the linear transformation T ( at 2 + bt + c ) = a b c a + b + c an isomorphism between P 2 and R 2 × 2 ? d) Let S be an invertible 3 × 3-matrix and A be a 3 × 3-matrix. Let T ( x ) = Sx be the linear transformation from Ker ( A ) to Ker ( SAS - 1 ). Is T an isomorphism? 2
3. (20) Answer the following questions. Give your reasoning. a) For what values of a, b, c is - 3 2 a 1 6 b 0 0 c the matrix of an orthogonal linear transformation?

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