Please show work TO 27.0 5. Consider the vectors p = 2 - x + 3x?,...
Q6 (4+3+3+ 6=16 marks) Let Xo, X1, X2 be three distinct real numbers. For polynomials p(x) and q(x), define < p(x),q(x) >= p(xo)q(x0) + p(x1)q(x1) + p(x2)q(22). Let p(n) denote the vector space of all polynomials with degree more no than n. (i) Show that < .. > is an inner product in P(2). (ii) Is < ... > an inner product in P(3)? Explain why. (iii) Is <,:> an inner product in P(1)? Explain why. (iv) Consider Xo =...
How to solve all of this linear Algebra 8. (24 points total) LetV be the vector space{P2, +, *}with standard function addition and scalar multiplication Define an Inner product: <p | q>= p(0)q[O) + p(1)q(1)+ p(2)q(2). Let B = {x,x,1} a. Explain why this inner product satisfies the positive property b. Explain how you know that B forms a basis c. State the conclusions of Cauchy-Schwartz and the Triangle inequalities in terms of this inner product d. Use Gram-Schmidt and...
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!
1. If the vectors and are orthogonal with respect to the weighted inner product < > = , what must be true about the weights ? 2. Do there exist scalars k and m such that the vectors p1 = 2+kx+6, p2 = m+5x+3 and p3 = 1 + 2x + 3 are mutually orthogonal with respect to the standard inner product on P2? N 12 We were unable to transcribe this image= 1211 + Աշշ W1, W2 We were...
Let Ps have the inner product given by evaluation at -2, -1, 1, and 2. Let po(t)-1. P,()-t, and p20)- a. Compute the orthogonal projection of p2 onto the subspace spanned by Po and P1 b. Find a polynomial q that is orthogonal to Po and p,, such that Po P is an orthogonal basis for Span(Po P1, P2). Scale the polynomial q so that its vector of values at a2(Simplify your answer.) Let Ps have the inner product given...
Let {p0, p1, p2} be a basis for a subspace V of ℙ3, where the pi are given below, and let the inner product for ℙ3 be given by evaluation at 0, 1, 2, 3, so <p,q> = p(0)q(0)+p(1)q(1)+p(2)q(2)+p(3)q(3). Use the Gram-Schmidt process to produce an orthogonal basis {q0, q1, q2} for V and enter the qi below. p0 = x−1 p1 = x2−2x+2 p2 = −3x2+2x q0 = q1 = q2 =
5.4.3. Consider the following set of three vectors. X2? 0 (a) Using the standard inner product in 4, verify that these vec tors are mutually orthogonal (b) Find a nonzero vector x4 such that (x1, x2, x3, x4) is a set of mutually orthogonal vectors. c) Convert the resulting set into an orthonormal basis for .
Can you help me? This is linear algebra. 3. (6) Let B-(1-3r,x +2x2,1-3x-8x2,2+x-5x2) be the set of vectors in P a) Is the set B a basis for P2? Justify. If it is not a basis for P, then extend B to a basis for P2 Calculator is allowed b) Use the basis found in part (a) to find the coordinate vector of f--1-3x-5x2 Calculator is allowed 3. (6) Let B-(1-3r,x +2x2,1-3x-8x2,2+x-5x2) be the set of vectors in P a)...
Consider a subset alpha={x+x2,1+x2,1 2x+2x2}ofP2(R). (a) Show that alpha is a basis for P2(R). (b) For f(x) = 1 + x + x2 2 P2(R), find its coordinator vector [f] alpha with respect to alpha. (c) Let = {1, x, x2} be the standard basis for P2(R), and let f(x) = a + bx + cx2 and g(x) = p+qx+rx2 be the elements of P2(R) and k 2 R. Prove that [f+g] = [f] +[g] and [kf] = k[f] and...
Please help for Question 10A.1 MATH 270 SPRING 2019 HOMEWORK 10 10A. 1. Let S be the subspace in R3 spanned by21.Find a basis for S 2. Using as the inner product (5) ( p. 246) in section 5.4 for Ps where x10, x2 -1, x3 - 2: Find the angle between p (x) = x-3 and q(x) = x2 + x + 2. b. Fnd the vector projection of p(x) on q(x) In Cl-π, π} using as an inner...