2. (a) Show that the list {p o(x),p 1(x), p 2(x),... ,P n(x),...,} wherePo(x) = 1, p1(x) = 1+x,P;(x) = 1+x ++ x³,•..)is a basis of the vector space F[x] of all polynomials with coeficients in F.(b) What are the coordinates of the vector x" relative to the basis (po(x),P1(x),..., P; (x),...)?

Answered: Image /qna-images/answer/ba29158c-e6f7-4b51-8890-e2401fc4f2ca.jpg

Answered: (a) Show that the list {p o(x), p 1(x),…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 25CM: Find a basis B for R3 such that the matrix for the linear transformation T:R3R3,...

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Similar questions

Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
Find a basis for R3 that includes the vector (1,0,2) and (0,1,1).
Let P2 be the polynomials of degree no more than 2. Is {x2+x+1,2x+1,3x2+1} a basis for P2 ?
(1) Show that the set S = {x, x + 1, (x + 1)^2} is a basis for the vector space P3 ofpolynomials of degree at most 2. (2) What are the coordinates of the polynomial x^2 + x + 1 with respect to the basis given in part 1?
. Let f1(x) = − 1 6 (x − 1)(x − 2)(x − 3), f2(x) = 1 2 x(x − 2)(x − 3), f3(x) = − 1 2 x(x − 1)(x − 3) and f4(x) = 1 6 x(x − 1)(x − 2). (i) Prove that B = (f1, f2, f3, f4) is a basis for P3[x], the vector space of all polynomials of degree at most three.
(a) Show that the vector functions are linearly independent on the real line. (b) Why does it follow from Theorem 2 that there is no continuous matrixP(t) such that x1 and x2 are both solutions of x' = P(t)x?
If f(x) = x^2 + 7x+2, g(x) = x^2 - x + 1 find a polynomial h(x) of degree 2 such that together these 3 are linearly independent Find constant c such that the three vectors (1,1,4),(2,1,1) and (7,5, c) DO NOT form a basis for 3d-space
Find a basis for the following subsapce of the vector space of polynomials of degree ≤ 3: W: = span {1+x+x2+x3, -1+5x+x2+11zx3, -1+2x+5x3, 1+6x+4x2+x3}
If q(x) is an arbitrary polynomial in Pn it follows that q(x) = c0p0(x) + ... + cnpn(x) (1) for some scalars c0, ... , cn (a) Show that ci = q ( ai) for i = 0, ... , n, and deduce that q(x) = q (a0)p0(x) + · · · + q( an)pn(x) is the unique representation of q(x) with respect to the basis ẞ
2-What is the size of the vector space consisting of polynomials of degree not exceeding n? A) 0 B) 2n+1 C) n-1 D) n+1 E) n
This is a textbook question, not a graded question The first four Hermite polynomials of the quantum oscillator areH0 = 1, H1 = 2x, H2 = 4x2 − 2, H3 = 8x3 − 12x. Show that these four polynomials form a basis H = {H0, H1, H2, H3} for the vectorspace P3 of polynomials of degree at most 3.

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