Math201fin1.pdf - B U Department of Mathematics Math 201 Matrix Theory Fall 2003 Final Exam This archive is a property of Bo\u02d8 gazi\u00b8ci University

Math201fin1.pdf - B U Department of Mathematics Math 201...

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B U Department of Mathematics Math 201 Matrix Theory Fall 2003 Final Exam This archive is a property of Bo˘ gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1.) Suppose A is a 4 × 3 matrix, and the complete solution to Ax = 1 4 1 1 is x = 0 1 1 + c 0 2 1 , c R (a) Find the second and the third columns of A . (b) Determine the ranks of the coefficient matrix and the augmented matrix. Give all the known information about the first column of A . Solution: (a) Let b T = [1 4 4 1]. From the particular solution when c = 0 it follows that column 2 + column 3 = b. The homogenous solution says that 2column 2 + column 3 = 0 . From these we obtain column 2 = - b, column 3 = b (b) We have dim Null( A ) = 1, which means rank( A ) = 2. Since the system is consistent, we have rank([ A | b ]) = rank( A ) = 2. Since the matrix must have two linearly independent columns, and all the remaining columns are multiples of b we can infer that the first column of A is not a multiple of b .
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2.) Let v 1 = 1 1 , v 1 = 1 4 , w 1 = 2 5 , w 2 = 2 8 and A = - 2 3 4 5 . (a) Show that β I = { v 1 , v 2 } is a basis for R 2 . Solution: Since dim R 2 = 2, any pair of linearly independent vectors form a basis for R 2 . The vectors v 1 and v 2 are linearly independent because 1 1 1 4 = 3 and therefore β I is a basis for R 2 . (b) Suppose A = [ T ] β I is the matrix representation of the linear transform T : R 2 R 2 relative to the basis β I . Find the matrix representation of the same linear transformation B = [ T ] β II with respect to the basis β II = { w 1 , w 2 } . Solution: If we can find a matrix M such that [ w 1 w 2 ] = [ v 1 v 2 ] M then B is related to A by B = M - 1 AM . We note that w 1 = v 1 + v 2 and w 2 = 2 v 2 , so M = 1 0 1 2 is the desired matrix. M - 1 = 1 0 - 1 2 1 2 B = M - 1 AM = 1 6 4 2 .
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3.) Let A = cos θ sin θ sin θ 0 . (a) Using the Gram-Schmidt process, find the A = QR factorization of A . (b) Find the projection matrix which projects onto the column space of A . Solution: (a) Let a = cos θ sin θ and b = sin θ 0 . Since a = 1 we set q 1 = a . Then we have b = b - ( q 1 T b ) q 1 and q 2 = b / b . We compute b = sin 2 θ sin θ - cos θ , q 2 = sin θ - cos θ . So Q = cos θ sin θ sin θ cos θ and R = q T 1 a q T 1 b 0 q T 2 b = 1 cos θ sin θ 0 sin 2 θ (b) If θ is not an integer multiple of π , then R has linearly independent columns, hence R T R is invertible, and P = A ( A T A ) - 1 A T = Q T Q = I = 1 0 0 1 . If θ is an integer multiple of π , then the second column of A is zero, and the first column is a = [1 0] T , so the column space is the one dimensional space spanned by a . The projection matrix onto this space is: P = aa T a T a = 1 0 1 0 = 1 0 0 0
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4.) (a) Using the cofactor matrix, find the inverse A - 1 of A = 1 1 1 1 2 2 1 2 3 .
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