Question 3. Let 3 5/' and for x(2),y -(,) ER2 define (a) Show that the assignment (x, y) > (x,y) defined ın (1) us an nner product [10 marks (b) If a - (1,-1) and b - (1,1), then show that the...
QUESTION 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3 (12) to find an orthogonal basis wth respect to the inner product defilned in question 2.1 for the span of this subset (2.3) Fınd all vectors (a, b, c) E R3 whuch are orthogonal to (1,0, 1) wnth respect to the...
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...
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,...
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...
5. (10 pts) Use the inner product < x,y > = 22191 +2242 in R2 and the Gram - Schmidt process to transform {(2, -1), (-2, 10)} into an orthonormal basis
Part Ill (10 pts each) 15. Let S {x2, (x- 1)2, (x -2)2 B) Define an inner product on P2 via < p(x) | q(x)>= p(-1)q(-1) p(0)q(0) +p(1)q(1) Using this inner product, and Gram-Schmidt, construct an orthonormal basis for P2 from S - use the vectors in the order given!
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
Problem 7: a) Show that the matrix A-I 5 we can use A to define an inner product onR2 by defining..erAir. b) With respect to this inner product, find an orthonormal basis for R2. c) With respect to this inner product, find the point in the span of closest to positive semidefinite Problem 7: a) Show that the matrix A-I 5 we can use A to define an inner product onR2 by defining..erAir. b) With respect to this inner product,...
5. For parts (a)-(d) below, consider the set of vectors B = {(1,2), (2, -1)}. (a) (2 points) Demonstrate that B is an orthogonal set in the Euclidean inner product space R2. (b) (3 points) Use the set B to create an orthonormal basis in the Euclidean inner product space R2 (e) (7 points) Find the transition matrix from the standard basis S = {(1,0),(0,1)} for R2 to the basis B. Show all steps in your calculation. (d) (7 points)...
Question 1 (2+2+5 marks] (a) Find the angle between the vectors y =(4,0,3), v = (0,2,0). (b) Consider the subspace V (a plane) spanned by the vectors y, V. Find an orthonormal basis for the plane. (Hint: you may not need to use the full Gram-Schmidt process.) (c) Find the projection of the vector w=(1,2,3) onto the subspace Vin (b). Hence find w as a sum of two vectors wi+w, where w, is in V and w, is perpendicular to...