I need proof of part (c) of the given theorem. 2.6-10 Theorem (Inverse operator). Let X, Y be vector spaces, bothreal or both complex. Let T: D(T)→Y be a linear operator withdomain D(T)cX and range R(T)c Y. Then:(a) The inverse T!: R(T)→D(T) exists if and only ifTx =0x = 0.(b) If T' exists, it is a linear operator.(c) If dim D(T) = n<∞ and T' exists, then dim R(T) = dim D(T).

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(c) If dim D(T) =n<∞ and T' exists, then dim R(T)=dim 2(T).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.2: Norms And Distance Functions

Problem 33EQ

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Ratios

A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.

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Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.

Question

I need proof of part (c) of the given theorem.

2.6-10 Theorem (Inverse operator). Let X, Y be vector spaces, both
real or both complex. Let T: D(T)→Y be a linear operator with
domain D(T)cX and range R(T)c Y. Then:
(a) The inverse T!: R(T)→D(T) exists if and only if
Tx =0
x = 0.
(b) If T' exists, it is a linear operator.
(c) If dim D(T) = n<∞ and T' exists, then dim R(T) = dim D(T).

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