6.1(3,8,11,17) 6.2(2,4,6)

6.1(3,8,11,17) 6.2(2,4,6) - Math 235 Homework 11 Solutions...

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Unformatted text preview: Math 235 Homework 11 Solutions April 14, 2004 ยง 6.1, Problem 3 Let f ( t ) = t and g ( t ) = e t . h f, g i = Z 1 te t dt = 1 . h f, f i = Z 1 t 2 dt = 1 / 3 . h g, g i = Z 1 e 2 t dt = e 2 / 2 โˆ’ 1 / 2 . h f + g, f + g i = Z 1 ( t + e t ) 2 dt = e 2 / 2 + 11 / 6 . ยง 6.1, Problem 8 Give reasons why each of the following is not an inner product on the given vector spaces. (a) On R 2 define h ( a, b ) , ( c, d ) i = ac โˆ’ bd . This fails to be an inner product because h (1 , 5) , (1 , 5) i = 1 โˆ’ 25 = โˆ’ 24 6 > . (b) On M 2 ( R ) define h A, B i = tr( A + B ). This fails to be an inner product space because โˆ’ 5 โˆ’ 5 , โˆ’ 5 โˆ’ 5 = tr โˆ’ 10 โˆ’ 10 = โˆ’ 20 6 > . 1 Math 235 Homework 11 Solutions April 14, 2004 ยง 6.1, Problem 11 Prove the parallelogram law for inner product spaces. || x + y || 2 + || x โˆ’ y || 2 = h x + y, x + y i + h x โˆ’ y, x โˆ’ y i , = h x, x + y i + h y, x + y i + h x, x โˆ’ y i โˆ’ h y, x โˆ’ y i , = h x + y, x i + h x + y, y i + h x โˆ’ y, x i โˆ’ h x โˆ’...
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This note was uploaded on 01/26/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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6.1(3,8,11,17) 6.2(2,4,6) - Math 235 Homework 11 Solutions...

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