5) In C.), with inner product <f,g> [f(x)g(x)dx, let f(x) = x², g(x)= x', a) Compute<...
5) In Coul, with inner product < f g >= $(x)g(x)dx, let f(x) = x”,g(x) = x, a) Compute< x,x?>; b) Find the "angle" between the two functions.
5) In Goy), with inner product < 1.8>]s«g(x)dx, let S(x)=x"$(x)=x', a) Computer />; b) Find the "angle" between the two functions.
(1 point) Use the inner product 1 0 <fig >= f(x)g(x)dx in the vector space Cº[0, 1] to find the orthogonal projection of f(x) = 6x2 + 1 onto the subspace V spanned by g(x) = x – į and h(x) = 1. projy(f) =
1.(16) Let P be an inner product space with an inner product defined as <.g > Ox)g(x)dx a) Let / =1+x.8=-2+x-x. Compute: <.8 >. The angle between / and g, and proj, b) Let h=1+ mx' in P Find m such that and h are orthogonal c) Let B = (1+x.I-XX+X' is a basis for P. Use the Gram-Schmidt process to covert B to an orthogonal basis for P. 2. Suppose and ware vectors in an inner product space V...
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
QUESTION 1 Let V-L2([0,1 ],C) and > : Vx-СУч . Г f(x)g(x)dx be an inner product on V Let gor 91, 92, 93:0,1]R be given by gox)-1,g1(x)-x, 920x)-x2, g3(x) -x3 and consider the following subset S = { go, g 1, g 2, g3JC V. After applying the Gram-Schmidt process the following set of vectors T = {vo, vľ, V2, V3} is an orthonormal set, where V1, V2, V3, and V4 are given by: O vo= 1, v,-V3(2x-1), v,-V5 (6x2-6x...
Problem 5. Let a < b and c > 0 and let f be integrable on [ca, cb]. Show that f c Ca where g(a) f(ex)
PROBLEM 1 Let the joint pdf of (X,Y) be f(x, y)= xe", 0<y<<< a. Compute P(X>Y). b. What is the conditional distribution of X given Y=y? Are X and Y independent? c. Find E(X|Y = y). d. Calculate cov(X,Y).
7 Consider the inner product space Co. 11 with the inner product defined by < 2,9 >= ( ( (x) g(x) dx (a) Show that f(x) = 1 and g(x) = 2x - 1 are orthogonal (b) Find ||g(2)|| (e) Find the distance d(f(x), g(x)) between f(x) and g(x)
Joint pdf is given for 0 SX < 2 and 0 sy 51 f(x,y) = 0.W. Find P(X+Y > 2).