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sample_tt2_2 - Math 235 1 Short Answer Problems Sample Term...

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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 Ax . e) For the matrix A = - 7 - 2 - 3 - 5 5 - 1 0 1 0 2 - 5 - 5 6 4 2 7 - 10 - 4 0 - 4 0 - 5 10 10 - 6 - 2 - 3 - 5 7 2 6 2 3 5 - 5 0 , - 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 . b) Consider A = 0 2 - 1 2 . Find the matrix B in S such that A - B is minimized.
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