sample_tt2_2 - Math 235 Sample Term Test 2 - 2 1. Short...

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Unformatted text preview: Math 235 Sample Term Test 2 - 2 1. Short Answer Problems a) State the Principal Axis Theorem. b) Let A be an m n matrix. Prove that A T A is symmetric. c) State the definition of a quadratic form Q ( ~x ) on R n being negative definite. d) Consider the quadratic form Q ( ~x ) = 3 x 2 + 5 y 2 + 3 z 2- 2 xy- 2 xz + 2 yz . Write down the symmetric matrix A such that Q ( ~x ) = ~x T A~x . e) For the matrix A = - 7- 2- 3- 5 5- 1 1 2- 5- 5 6 4 2 7- 10- 4- 4- 5 10 10- 6- 2- 3- 5 7 2 6 2 3 5- 5 ,- 1 is an eigenvalue with multiplicity three, and 2 is an eigenvalue with multiplicity two. Determine the determinant of A . 2. Let A = 4 2 2 2 4 2 2 2 4 . Find an orthogonal matrix P that diagonalizes A and the corresponding diagonal matrix. 3. On M (2 , 2) define the inner product < A,B > = tr( A T B ). a) Use the Gram-Schmidt procedure to produce an orthonormal basis for the subspace S = Span 1 0 0 1 , 0 1 0 1 , 1 0 1 1...
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sample_tt2_2 - Math 235 Sample Term Test 2 - 2 1. Short...

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