(4) Let C[0,1] be the inner produce space of all real-valued, continuous functions on the interval...
33. Let C[0, 1] be the space of real-valued continuous functions on [0, 1] with inner product Kf.9) (x)g()d(r). 2 cos 2Tir and g.(r)-v2 sin 2mix for i 1.2,... Show that (1. fi.g. f2 92 Suppose that fi(r) is an orthonormal set.
2) Let CI0,1] be the vector space of all continuous real valued functions with domain [0,1J.Let (f.8)-Co)ds be the inner product in C10.11 where fand g are two functions in CI0,1. Answer the following questions for f(x)-x and g(x)-cos. a) Find 《f4) and i g I where l.l denotes the length induced by this inner product,Show your work b) Determine the scalar c so that f-cg is orthogonal to f.Show all your work.
34. Let V be the subspace of the vector space of all real- valued continuous functions that has basis S = {e'. e-}. Show that V and Rare isomorphic.
advanced linear algebra, need full proof thanks Let V be an inner product space (real or complex, possibly infinite-dimensional). Let {v1, . . . , vn} be an orthonormal set of vectors. 4. Let V be an inner product space (real or complex, possibly infinite-dimensional. Let [vi,..., Vn) be an orthonormal set of vectors. a) Show that 1 (b) Show that for every x e V, with equality holding if and only if x spanfvi,..., vn) (c) Consider the space...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample (2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...
3. Let V-CỦ-π, π]), the vector space of continuous functions on [-π, π]. Let (a) Prove that ( , ) is an inner product (b) Let S-{sin r, cos z, sin 2r, cos 2r, sin 3x, cos 3x,...n-1,2,. Show that S is a set of orthonormal vectors 3. Let V-CỦ-π, π]), the vector space of continuous functions on [-π, π]. Let (a) Prove that ( , ) is an inner product (b) Let S-{sin r, cos z, sin 2r, cos...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
interval-1,1. If f.geCL1.], we'l 7) The field of play is Cil the space of all functions that are continuous on the define the inner product as (f,g)= f'f(x)g(x)dx. The question is simply this: Find the orthogonal projection of e" onto P, and graph both functions on [-2,2]. interval-1,1. If f.geCL1.], we'l 7) The field of play is Cil the space of all functions that are continuous on the define the inner product as (f,g)= f'f(x)g(x)dx. The question is simply this:...
e interval -1,1].if f.ge C[L.] 7 The field of play is C the space of all functions that are continuous on th we'll define the inner product as (f.g)= 5(x)g(x)dx. The question is simply this: Find the orthogonal projection of e onto P, and graph both functions on [-2,2]. e interval -1,1].if f.ge C[L.] 7 The field of play is C the space of all functions that are continuous on th we'll define the inner product as (f.g)= 5(x)g(x)dx. The...
Please attempt both questions 3. Use the continuous function on the interval (0,1) inner product to find the projection of f(x) onto g(x). (Feel free to use an integral calculator. I use wolfram alpha. Just make sure to type the problem in carefully). (a) f(x) = -22 - 1, g(x) = -2 (b) f(x) = 2r?, g(x) = 2+1 (e) f(x)=-1-1, g(x) = x2 +3 4. Consider 3-space with the dot product. Your subspace S will be the plane z...