5 5) Suppose DE L(R;[x], R2[x]) is the differentiation map defined by Dp = p'. RecallF,[x] is the vector space of all polynomials over a field F of degree at most n and has astandard basis of {1, x, x²,...,x"}. Find a basis of R3[x] and a basis of R2[x] such that'1 0 0 0the matrix of D with respect to these bases is ( 0 1 00 0 10.6) Define the linear transformation T: R 3by T

Answered: Image /qna-images/answer/6578485c-73c3-4a57-ae7b-c913b82b6767.jpg

Answered: F,[x] is the vector space of all…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

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Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..
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.
Given that f is a polynomial of degree n in Pn, show that { f, f ',f ", . . . , f (n)} is a basis for Pn. ( f (k) denotes the kth derivative of f.)
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
(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 T: V-->W be a linear transformation between finite-dimensional vector spaces V and W Let B and C be bases for V and W, respectively, and let A= [ T]C<--B. Use the results of this section to give a matrixbased proof of the Rank Theorem
(a) Prove that the set ẞ = {p0(x), p1 (x), ... , Pn(x)} of Lagrange polynomials is linearly independent in Pn (b)Deduce that ẞ is a basis for Pn
Let T: V-->W be a linear transformation between finite-dimensional vector spaces V and W Let B and C be bases for V and W, respectively, and let A= [ T]C<--B. Show that rank(T) = rank(A).

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F,[x] is the vector space of all polynomials over a field F of degree at most n and has a standard basis of (1, x, x2,.,x"}. Find a basis of R,[x] and a basis of R2[x] such that 1 00 0 the matrix of D with respect to these bases is (0 1 0 0 0. 6) Define the linear transformation T: hy T