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Unformatted text preview: Math 235 - Final Exam Fall 2009 NOTE: The questions on this exam does not exactly reflect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not asked here. 1. Short Answer Problems a) Let A = 3 + i 2 i 1 . Compute A * . b) Let A = 3 i- i 2 . Is A normal? c) Show that the product of two orthogonal matrices is an orthogonal matrix. d) Let A = 3 0 0 2- 1 0 1 2 . Find an orthogonal matrix P such that P T AP is in real canonical form. 2. Let W be the vector space of all 2 2 upper triangular matrices with real entries. So W = a b c | a,b,c R . Consider the linear mapping L : R 3 W defined by L ( x,y,z ) = x + y x + z y- z . a) Find the rank and nullity of L . b) Explain why L is not an isomorphism. Explain why R 3 and W are isomorphic without finding an isomorphism between them....
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- Spring '08