CHE1)A) Let V and w be vector spaces such that dim (V) = dim (W)and let T:V-aw be linear. Show there exists ordered bases B and IPor V and w, respectively, such that [T] is a diagonal matrix.(Hint: dimensio theorem proof).3B) Consider the basis x= [('0°), (b), (i), (0°)) of Mara (F).Let T: M₂xz (F) —> Maxz (F) defined by TCA) = At. Use theorem 2.14to compute [TCA)] where A = (-₁)Theorem 2.14B = [v₁₁.., vn} x = {W,,.., wn). IF T:V-> W linear, then [T(v)]x.[T]• [V] = [T(V)]

Answered: на 1) A) Let V and w be vector spaces…

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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Question

C
HE
1)
A) Let V and w be vector spaces such that dim (V) = dim (W)
and let T:V-aw be linear. Show there exists ordered bases B and I
Por V and w, respectively, such that [T] is a diagonal matrix.
(Hint: dimensio theorem proof).
3
B) Consider the basis x= [('0°), (b), (i), (0°)) of Mara (F).
Let T: M₂xz (F) —> Maxz (F) defined by TCA) = At. Use theorem 2.14
to compute [TCA)] where A = (-₁)
Theorem 2.14
B = [v₁₁.., vn} x = {W,,.., wn). IF T:V-> W linear, then [T(v)]x.
[T]• [V] = [T(V)]

Expert Solution

Introduction

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Since you have posted multiple questions, we will provide the solution only to the first question as per our Q&A guidelines. Please repost the remaining questions separately.

In linear algebra, one important concept is that of diagonalizability of a linear transformation. As per the question we are given two vector spaces V and W of equal dimension and a linear transformation T : V → W, we aim to show that there exist ordered bases B and γ for V and W, respectively, such that the matrix representation [T]_B^γ is a diagonal matrix. In other words, we want to show that T can be diagonalized.