Let CTO,1] 3.(12 pts) Let f=f(x)-mx-1 and g = g have the inner product (x)-4x +...
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...
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 C.), with inner product <f,g> [f(x)g(x)dx, let f(x) = x², g(x)= x', a) Compute< x², x? >; 0 b) Find the “angle” between the two functions.
Linear Algebra 2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
Let f(x)=2x² – 7 and let g(x) = 4x + 1. Find the given value. f(g(-3)] f(g(-3)) = (Type an integer or a decimal) Question Viewer
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,...
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...
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] →...
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.
(3 points) The function (f, g) = f(-2x(-2) + f(0)g(0) + f(2)g(2) defines an inner product on P2. With respect to this inner product, find the orthogonal projection of /)-4x2 +5x-3 onto the subspace L spanned by g(x) = 2x2-2x-4.