e3-sample - ~x ~ y = k ~x + ~ y k 2-k ~x-~ y k 2 2 . Use...

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Math 204. Exam 3 (practice) November 17th, 2010 This is the practice exam. The actual exam is about the same length. Note that this is supposed to give you a feeling for what the actual exam will be like. It does not represent every “type” of problem. 1. Prove that the orthogonal projection of a vector ~x onto the subspace V is the point in V that is closest to ~x . 2. Suppose that P is a linear transformation and that P 2 = P = P T . Show that there is a subspace V such that P is the orthogonal projection onto V . In addition prove that I - P is the projection onto V . 3. Give examples of the following: (a) A matrix that is not symmetric. (b) A 2 × 2 orthogonal matrix Q such that Q~e 1 = ~ e 1 + ~ e 2 2 . (c) A matrix A such that A 2 = A , but A 6 = A T . 4. Find the minimal energy solution for the system A = ± 1 0 1 0 1 - 1 ² ~ b = ± 3 2 ² . 5. Show that rank( A ) = rank( A T ). Show that ker( A ) = ker( A T A ) 6. The determinant of the matrix A = [ ~v 1 |···| ~v 4 ] is 5. Compute the determinant of the matrix B = [ ~v 3 | ~v 1 + ~v 2 | ~v 1 | 2 ~v 4 ] 7. Compute the area of the triangle with vertices (1 , 2), (3 , 4) and ( - 1 , 6). 8. Show that
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Unformatted text preview: ~x ~ y = k ~x + ~ y k 2-k ~x-~ y k 2 2 . Use this fact to prove that a linear transformation that preserves length, must also preserve angles. 9. If A R n n is a matrix, then show that A + A T is symmetric. A matrix H is called skew-symmetric if H T =-H . Show that A-A T is skew-symmetic. Use this to prove that every matrix A can be written in the form A = B + C where B is symmetric and C is skew-symmetric. 10. (hard) The simplex spanned by ~v 1 ,...,~v n R n is the set of vectors of the form t 1 ~v 1 + ...t n ~v n , where t 1 ,...,t n [0 , 1]. The convex hull is the set of points of the form t 1 ~v 1 + + t n ~v n where t 1 + + t n = 1. 1 (a) In two dimensions what is the shape of the simplex and the convex hull of two linearly independent vectors. What is the relationship between their areas. (b) Generalize your answer to n dimensions. 2...
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This document was uploaded on 10/29/2011 for the course MATH 204 at Vanderbilt.

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e3-sample - ~x ~ y = k ~x + ~ y k 2-k ~x-~ y k 2 2 . Use...

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