Assignment 8

Assignment 8 - = Z 2- 2 f ( x ) g ( x )cos xdx denes an...

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MATH 223, Linear Algebra Winter, 2008 Assignment 8, due in class Wednesday, March 19, 2008 1. Calculate the following; the inner product space is C 3 , with the standard inner product. < 1 1 - i 3 - 2 i , - 2 i 3 - 7 i 4 - i > , < 1 - i 2 + i 3 , 1 - i 2 + i 3 > , < 2 + i 1 - i 0 , 2 - i 1 + i 0 > . 2. Suppose that A is a Hermitian matrix and that it has an eigenvalue λ 0. Show that A is not positive definite; that is, there is a nonzero vector ~v such that ~v T A ~v 0. 3. For each of the following matrices A j , decide whether the function on C 3 × C 3 defined by < ~v, ~w > = ~v T A j ~w is an inner product on C 3 . A 1 = 1 1 1 1 1 1 1 1 1 , A 2 = 1 1 - i 2 + 3 i 1 - i 3 5 i 2 + 3 i 5 i 0 , A 3 = 6 0 1 + i 0 3 0 1 - i 0 2 . 4. Let V = C [ - π 2 , π 2 ], the real vector space of continuous function on the closed interval [ - π 2 , π 2 ] . (a) Verify that < f,g >
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Unformatted text preview: = Z 2- 2 f ( x ) g ( x )cos xdx denes an inner product on V . (b) Verify that < f,g > = Z 2- 2 f ( x ) g ( x )sin xdx does not dene an inner product on V . 5. Let V = R [ x ], the real vector space of polynomials with real coecients. We dene < f,g > = R 1-1 f ( x ) g ( x ) dx , the standard inner product on this space. (a) Find a scalar a such that < a,a > = 1. (b) Find scalars b and c such that < a,b + cx > = 0 and < b + cx,b + cx > = 1. (c) Find scalars , and such that < a, + x + x 2 > = < b + cx, + x + x 2 > = 0 and < + x + x 2 , + x + x 2 > = 1 . 1...
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This note was uploaded on 04/29/2008 for the course PHYS 230 taught by Professor Harris during the Fall '07 term at McGill.

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