Recall that M2,2 is the vector space of real 2 x 2-matrices and P3 is the vector space of real polynomials ofdegree at most 3. Let T: M2,2 → P3 be a linear function satisfyingFind the following:( [([(a) T(b) T(c) T(d) T(e) T10000001==(12 7³])=17(D)--+= -4,T0T( )=-*- 3.3,T([i])=-4x². T([8])=3x² - 4x::

Answered: Image /qna-images/answer/83672a7c-2375-4a9f-a372-7f3c8f8557f5.jpg

Answered: Recall that M2,2 is the vector space of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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In the vector space P3(R), let S = {p1(x), P2(x), P3(x)}, where p1 (x) = 63x° + 19x² – 37x – 41, P2(x) = –189x – 57x² + 111x + 123, P3 (x) = 45x – 26x? – 55x + 92. (a) Determine whether S is linearly independent. (b) Determine whether (S) = P3(R). (c) Find the dimension of (S). 4.
Let V = {(x, 8) | x E R}. You are given that (V, 0, O) is a vector space with (x1, 8) O (x2, 8) = (x1+ x2, 8) and c O (x, 8) = (ca, 8). What is the zero element of V? What is the additive inverse of (5, 8) in V?
Let V be the vector space of functions spanned by B = {b₁ = 2 sin x + 9 cos x, b₂ = 4 sin x + 7 cos x} where x C = {C₁ = sin x, C₂ = cos } where x of coordinates matrix P. C+B P = C←B [f(x)] c = [ The coordinates of a function f(x) relative to the basis B are [f(x)] B = [8] 6 coordinates of f(x) relative to C and find f(x). f(x) = Ex: 3 Ex: 3 nn 2 sin x + -, ne Z nn 2,n € Z is also a basis for V. Find the change COS x Find the
Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D : P3 - P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) = P3. Is D a linear transformation? Let p(x) = a³x³ + ²x² + a₁ + a₁ and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = = Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ✪ c. Is D a linear transformation? choose ◆
T: P2 P2 defined by T(a + bx + cx²) = a + b(x + 8) + b(x + 8)² Find the following images of the function for vectors p₁(x) = a₁ + b₁x + c₁x² and p₂(x) = a₂ + b₂x + c₂x2 in P₂ and the scalar k. (Give all answers in terms of a₁, b₁, C₁, 22, b₂, C₂ and c.) T(P₁) + T(P₂): = T(P1 + P₂) = cT (P1) = T(CP1) = Determine whether T is a linear transformation. 4 Olinear transformation O not a linear transformation
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
Let d : P5(R) → P5(R) denotes the first derivative. Show that d is a linear map. Choose a basis for P5(R) and find the matrix M(d). Use the matrix M(d) to find the derivative of f(x) = −9x^5 + 11x^4 −4x^2 +5x−1.
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.

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