1.(16) Let P be an inner product space with an inner product defined as <.g >...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be defined by ull , ).Consider the target function f (x) with the approximating space P e', and work 2. Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials p (x) through degree four. Standardize your polynomials such that p, (1) 1 (b) Find the best degree 4 approximation to f(x) using the specified norm, and working with this...
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)
(1 point) Consider the complex inner product space C with the usual inner product Let -4i 4i and let w = span(vi,V2). (a) Compute the following inner products: (v.vi)- 2 (Vi, V2-12 (2. V)12 (b) Apply the Gram-Schmidt procedure to Vi and v2 to find an orthogonal basis (ui,u2l for W , u2=1112
QUESTION 8 Let (V,<,>) be an inner product space, and P: V – V a linear map. Choose the correct statement(s). Multiple choices might be correct and wrong choices have negative points. if P(V) = < W, V > Wand ||w|= 1, then P is an orthogonal projection. if P is an orthogonal projection, then < V- P(V), W> = 0 for any VEV, welmP. fW= Im P and {W 1,...,Wx} is an orthonormal basis for W then P(V) =...
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...
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,...
3. Consider the inner product space V = M2x2(C) with the Frobenius inner product, and let T:V + V be the linear operator defined by T(A) -1 (i) Compute T:((1+i :)) (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,) in such a way that the induced metric is complete. In particular, there is a norm on X defined by and the metric is given by d(r, y) yl Let A denote the unit ball A x E X < 1} We know that A is closed and bounded essentially from the definitions. Show that A is not compact. (Hint: Construct a sequence xn...
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i - 6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...