a) Find an orthonormal basis for the subspace of R₄ consisting of all vector of the form [ a a+b c b+c ].

ans given: [1/√2 1/√2 0 0 ], [-1/√6 1/√6 0 2/√6], [1/√12 -1/√12 3/√12 1/√12 ]

b) Let S={t+1,t-1} be a basis for a subspace W of the Euclidean space P_{2 .} Find an orthonormal basis for W.

ans given: {√(3/7) *(t+1), (1/√7)*(9t-5) }

c) [note S is a set of 2 x 2 matrix where first 2 numbers are on top and last 2 on bottom]Let S={ [0 0 0 1], [1 1 0 0], [1 0 01] } be a basis for a subspace W of the Euclidean space of all 2 x 2 matrices with inner product definded by (A,B)=Tr(B^TA). Use the Gram-Schmidt process to find an orthonormal basis for W.

ans given: [0 0 0 1], 1/√2 [1 1 0 0], 1/√2 [1 -1 0 0]

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