7. Let V = Pa(R), the vector space of polynomials over R of degree less than 2, with inner product Define φ E p by φ(g)-g(-1) a) By direct calculation, find f e V such that (S)-dg). You are given...
e the vector space of polynomials over R of degree less than 3. Define a quadratic form on V by a) Find the symmetric bilinear forma f such that q(p) = f(p, p). b) Consider the basis oy-(1,2-x U)o. c) Let R-(3,2-r, 4-2z +2.2} of V. Find the matrix {f}3: You may give your ,24 of V. Find the matrix answer as a product of matrices and/or their inverses. e the vector space of polynomials over R of degree less...
let P3 denote the vector space of polynomials of degree 3 or less, with an inner product defined by 14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements 14. Let Ps denote the vector space of polynomials of degree 3 or less,...
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)?...
Given the vector space R[2]deg<s of polynomials with real coefficients of degree at most 5, and Ui = {p(z) : p(z) a? + bz5, for abe R}, find a subspace U2 such that R deg< 5 = Ui φ Ủy Is this U2 unique? (g) If V be a finite dimensional vector space, dim V = n and B = 〈ui,u2, . . . , un) is a basis of V, then show that:
2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг + dr + ey + f | a,b,c, d, e, f E R} Let T be the linear operator on V defined by Find the Jordan canonical form of T 2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг...
2 over R. Define U g-Ло íj fg for f,0€ y 5. Let V be the vector space of polynomials of degree a. If U is the subspace of scalar polynomials, find U b. Apply Gram-Schmidt to the basis1, t,t2) of V. c. If a E R find ga E V with (f, ga)(a) for all f E V
Let V be the vector space of all polynomials of degree at most 2 equipped with the inner product defined by (p,q) = p(-1)q (-1) + p (0)g(0) +p(1)q(1),p(x),g(x) E V Find a nonzero polynomial that is orthogonal to both p(x) = 1 + x + x2, and q(x) = 1-2x + x2
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f). (7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
6. (a) Let V be a vector space over the scalars F, and let B = (01.62, ..., On) CV be a basis of V. For v € V, state the definition of the coordinate vector [v]s of v with respect to the basis B. [2 marks] (b) Let V = R$[x] = {ao + a11 + a222 + a3r | 20, 41, 42, 43 € R} the vector space of real polynomials of degree at most three. Write down...
NEED (B) AND (C) 2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of continuous real-valued funo- tions on the domain [-1, 1] (b) Use the Gram-Schmidt process to find an orthonormal basis for P2(R) with re- spect to this inner product (c) Find a polynomial q(x) such that for every p E P2R 2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space...