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sample_tt2_1_ans

# sample_tt2_1_ans - Math 235 Sample Term Test 2 1 Answers...

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Math 235 Sample Term Test 2 - 1 Answers NOTE : - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Let S be a subspace of an inner product space V . What is the deﬁnition of S . Solution: S = { ~x V | < ~x,~s > = 0 for every ~s S } . b) State the Principal Axis Theorem. Solution: Let A be an n × n matrix. A is symmetric if and only if A is orthogonally diagonalizable. c) Determine the matrix for the quadratic form Q ( x, y, z ) = 3 x 2 - y 2 + z 2 - 2 xy + 2 yz. Solution: A = 3 - 1 0 - 1 - 1 1 0 1 1 . d) Write the deﬁnition of a quadratic form Q on R n being positive deﬁnite. Solution: Q is positive deﬁnite if Q ( ~x ) > 0 for all ~x R n , ~x 6 = ~ 0. 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 . Solution: k ~v k = 5. 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. Solution:

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sample_tt2_1_ans - Math 235 Sample Term Test 2 1 Answers...

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