# PS1 - of orthogonal functions g i in terms of the f i 7...

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MATH 580: Problem Set 1 due Tuesday, September 14, 2004 1. Are y 2 and y 4 linearly independent? Why or why not? 2. Keener 1.1.3a (p.49) 3. Keener 1.1.5 (p.49) 4. Keener 1.1.8 (p.49) 5. Why isn’t || x || ≡ h x, x i a valid induced norm? 6. Let f 1 , f 2 , f 3 , . . . be a set of linearly independent functions in the space L 2 [ a, b ], which means functions square-integrable on the interval [ a, b ], i.e. Z b a [ f ( x )] 2 dx < Using the Gram-Schmidt orthogonalization process, deﬁne a sequence
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Unformatted text preview: of orthogonal functions g i in terms of the f i . 7. Using the generalized deﬁnition of the angle between two elements of a inner product space, (a) show that the angle between x and x 2 is about 14 . 5 ◦ , where the inner product h f, g i ≡ Z 2 f ( x ) g ( x ) dx ; (b) what would the angle between the same two functions be for the innner product h f, g i ≡ Z 6 1 f ( x ) g ( x ) dx...
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## This note was uploaded on 07/23/2008 for the course MATH 580 taught by Professor Belmonte during the Fall '04 term at Penn State.

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