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Unformatted text preview: ~x Â· ~ y = k ~x + ~ y k 2k ~x~ y k 2 2 . Use this fact to prove that a linear transformation that preserves length, must also preserve angles. 9. If A âˆˆ R n Ã— n is a matrix, then show that A + A T is symmetric. A matrix H is called skewsymmetric if H T =H . Show that AA T is skewsymmetic. Use this to prove that every matrix A can be written in the form A = B + C where B is symmetric and C is skewsymmetric. 10. (hard) The simplex spanned by ~v 1 ,...,~v n âˆˆ R n is the set of vectors of the form t 1 ~v 1 + ...t n ~v n , where t 1 ,...,t n âˆˆ [0 , 1]. The convex hull is the set of points of the form t 1 ~v 1 + Â·Â·Â· + t n ~v n where t 1 + Â·Â·Â· + 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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 Fall '08
 STAPLES
 Math, Linear Algebra, Algebra, Vector Space, orthogonal projection

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