A5_soln.pdf - Math 235 Assignment 5 Solutions 1 Let S =...

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Math 235 Assignment 5 Solutions 1. Let S = Span { 1 + x 2 } be a subspace of P 2 ( R ). Find a basis for S under the inner product h p, q i = p ( - 1) q ( - 1) + p (0) q (0) + p (1) q (1). Solution: Every vector a + bx + cx 2 S satisfies 0 = h 1 + x 2 , a + bx + cx 2 i = 2( a - b + c ) + 1( a ) + 2( a + b + c ) = 5 a + 4 c Thus, a = - 4 5 c , so every vector in S has the form a + bx + cx 2 = - 4 5 c + bx + cx 2 = bx + c - 4 5 + x 2 Hence, a basis for S is { x, - 4 + 5 x 2 } . 2. Let S = Span 1 - 1 1 , 1 0 1 be a subspace of R 3 and let ~w = 1 1 2 . a) Find an orthonormal basis for S .
b) Find proj S ~w . Solution: proj S ~w = ( ~w · ~w 1 ) ~w 1 + ( ~w · ~w 2 ) ~w 2 = 2 3 1 / 3 - 1 / 3 1 / 3 + 5 6 1 / 6 2 / 6 1 / 6 = 3 / 2 1 3 / 2 c) Find a basis for S . Solution: Since dim R 3 = 3 and dim S = 2, we have that dim S = 3 - 2 = 1. Also, perp S ~w = 1 1 2 - 3 / 2 1 3 / 2 = - 1 / 2 0 1 / 2 S . Hence, - 1 / 2 0 1 / 2 is a basis for S .

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