Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Define an isomorphism from $P_{3}$ to $R^{3}$ .

The solution is not to define an isomorphism.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

$T (f + g) = T (f) + T (g) T (kf) = kT (f)$

For all elements $f, g$ of V and k is scalar.

A linear transformation $T : V \to W$ is said to be an isomorphism if and only if $\ker (T) = {0}$ and $im(T)=W$ or $\dim (V) = \dim (W)$ .

Step2: Explanation of the solution

Consider the transformation as follows.

$T : P_{3} \to R^{3}$

Since,role="math" localid="1659418414775" $\dim (P_{3}) = 4$ and $\dim (R^{3}) = 3$

Therefore, $\dim (P_{3}) \neq dm (R^{3})$ .

Any two finite dimensional spaces will be isomorphic if and only if they will have the same dimension.

Thus, there is not possible to define any isomorphism from $P_{3}$ to $R^{3}$ .

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Linear Algebra With Applications

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Short Answer

Define an isomorphism from $P_{3}$ to $R^{3}$ .

Step by Step Solution

Step1: Definition of Linear Transformation

Step2: Explanation of the solution

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Linear Algebra With Applications

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Short Answer

Define an isomorphism from P3 to R3.

Step by Step Solution

Step1: Definition of Linear Transformation

Step2: Explanation of the solution

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Define an isomorphism from $P_{3}$ to $R^{3}$ .