1. Consider the linear map T : R[X]deg<3 –→ Mat2,2(R) given byT(f) =f(0) f'(1)(f"(-1) f(2)$(2) ).Let B = {X, X², 1, X³} be a basis for R[X]deg<3 and-{( :). (; ?) - (: )- (: )}:=1 1a basis for Mat2,2 (R). Find [T]c+B, and determine whether T is injectiveand/or surjective.

Answered: Image /qna-images/answer/55910395-f046-4deb-ba3f-4bc6fd8f3cab.jpg

Answered: 1. Consider the linear map T :…

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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1. (Section 17.1) Match the vector fields with their sketches below by placing the letter of the function in the corresponding blank: (I) F(x, y) = (II) F(x, y) = (III) F(x, y) = (IV) F(x, y) = Vector Field: (B) Vector Field: (A) (C) Vector Field (D) 0 Vector Field:
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4
Let U,V ,W be finite-dimensional F-vector spaces and let T:U → V, S:V→W be linear transformations. Prove that nullity(S • T) - nullity(T ) = dim(im(T ) N ker(S)) following steps (a)-(c) below. a) LetU:=ker(S-T)andT:=T|u.Prove:im(T').=T(U')=im(T)N ker(S ). b) Prove:ker(T')=ker(T). c) Deduce:nullity(S•T)-nullity(T)=dim(im(T)nker(S)).
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1. Consider the linear map T : R[X]deg<3 → Mat2,2(IR) given by f(0) f'(1) T(f) = (-1) F(2) Let B = {X, X², 1, X³} be a basis for R[X]deg<3 and - {(' :) (: ) - (6 :). ( )} (6 ) · (: )} C : a basis for Mat22(R). Find [T]c-8, and determine whether T is injective and/or surjective.