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Math 304 Quiz 15 Summer 2007 Linear Algebra 1. Find values of 1 the matrix 2 3 is an eigenvector of a and b for which the vector 4 a with eigenvalue 5. b Solution. The given information says that 1 a 3 3 =5 . 2 b 4 4 Equivalently, 3 + 4a = 15, 6 + 4b = 20. Solving each of these equations shows that a = 3 and b = 7/2. R1 2. In the space C[0, 1] with inner product hf, gi = 0 f (x)g(x) dx, use the Gram-Schmidt procedure to find an orthonormal basis for the subspace spanned by the functions 1 and x. R1 Solution. Since k1k2 = h1, 1i = 0 12 dx = 1, the function 1 is alR1 ready normalized. Now hx, 1i = 0 x dx = 21 , so the function x − 12 is orthogonal to the function 1. It remains to normalize the function x − 12 . R1 R1 1 Since kx − 12 k2 = 0 (x − 21 )2 dx = 0 (x2 − x + 41 ) dx = 31 − 21 + 41 = 12 , √ √ 1 the normalized function equals 12 (x − 2 ), or 3 (2x − 1). √ Thus the required orthonormal basis is [1, 3 (2x − 1)]. June 26, 2007 Dr. Boas