In each pari, apply the Gram Schmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for span(S'). Then normalize the vectors in this basis to obtain an orthonormal basis ri for span(S'). and compute the Fourier coefficients of the given vector relative to 0. Finally, use Theorem 6.5 to verify your result. (a) V - R: \ S - {(1.0, 1).(0. I. I). (1.3.3)}, and x = (1,1,2)
(b) V - R;! . S = {(1, 1, I), (0,1, I), (0,0,1)}, and x - (1,0,1)
(c) V - P2(/?) with the inner product (f(x),g(x)) - J* f(t)g(t)dt, S= {\..r..r2 }. and h(x) = 1+X
(d) V - span(.S'). where S = {(1,2,0), (1 -1,2,4*)}, and x = (3 + i,4i, 1)
(e) V - R1 . S - {(2. 1,-2,4), (-2,1,-5,5), (-1,3,7,11)}, and x = (-11.8.-4.18)
(f) V = R4 , S= {(1,-2, -1,3), (3,6,3, 1), (1,4,2,8)}, and x = (-1,2,1,1) / \ \, K, /i,