sample_tt2_1 - Math 235 Sample Term Test 2 - 1 1. Short...

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Unformatted text preview: Math 235 Sample Term Test 2 - 1 1. Short Answer Problems a) Let S be a subspace of an inner product space V . What is the definition of S . b) State the Principal Axis Theorem. c) Determine the matrix for the quadratic form Q ( x, y, z ) = 3 x 2- y 2 + z 2- 2 xy + 2 yz. d) Write the definition of a quadratic form Q on R n being positive definite. e) Let { ~w 1 , ~w 2 , ~w 3 } be three orthonormal vectors in R 27 . Let ~v = ~w 1- 4 ~w 2 + 8 ~w 3 . Compute the length k ~v k of ~v . 2. Let A = 2 1 1 1 2 1 1 1 2 . Find an orthogonal matrix P that diagonalizes A and the corresponding diagonal matrix. 3. Let V be an n-dimensional vector space, and let h , i be an inner product on V . Let { ~v 1 , . . . ,~v k } be an orthonormal basis for a subspace S of V , let P = proj S : V V be the projection onto S . a) Given an expression for P ( ~v ). b) Show that P P = P ....
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