3. Let V-CỦ-π, π]), the vector space of continuous functions on [-π, π]. Let (a) Prove that ( , )...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
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.
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...
4. Let L2(-π, π)) be the Lebesgue space of square integrable functions f: [-π, π] → C with inner-product, (f,g) =| f(t)g(t)dt (a) Show thatkt k e is an orthonormal system 2rZ s an orthonormal system (b) Let M be the linear span of (1, et, e). Find the point in M closest to the function [4 marks] 2π f(t) = t. [6 marks] 4. Let L2(-π, π)) be the Lebesgue space of square integrable functions f: [-π, π] →...
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...
30. (Hard) (Fourier series) Show that the infinite set はァsin nz.lcos nz : n-1,2. . . .} is an orthonormal set in the vector space C10,2 equipped with the inner product of real continuous functions on the interval io,2π] 2π 30. (Hard) (Fourier series) Show that the infinite set はァsin nz.lcos nz : n-1,2. . . .} is an orthonormal set in the vector space C10,2 equipped with the inner product of real continuous functions on the interval io,2π] 2π
Prob 4. Consider a complex vector space Vspan (1, cos r, sinr, cos 2x, sin 2x) with an inner product fog@dt. Let U be the subspace of odd functions in V. What is U1? Find an orthonormal basis for both U and U Prob 4. Consider a complex vector space Vspan (1, cos r, sinr, cos 2x, sin 2x) with an inner product fog@dt. Let U be the subspace of odd functions in V. What is U1? Find an orthonormal...
DO NOT COPY INCORRECT ANSWERS FROM ONLINE !! WRITE IN YOUR WORDS !! This is an Advanced Linear Algebra Question. Please answer only if your answer is fully sure, Otherwise please don’t answer the question leave it for a capable personal. DO NOT COPY INCORRECT ANSWERS FROM ONLINE !! WRITE IN YOUR WORDS !! Prob 4. Consider a complex vector space V span (1, cos z, sin z, cos 2r, sin 2x) with an inner product Let U be the...
Let V be a finite-dimensional vector space, and let B be a basis of V. Show that there is an inner product on V for which B is orthonormal.