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Q63E

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Found in: Page 185

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Define an isomorphism from ${{\mathbf{P}}}_{{\mathbf{3}}}$ to ${{\mathbf{R}}}^{{\mathbf{3}}}$.

The solution is not to define an isomorphism.

See the step by step solution

## 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.

${\mathbf{T}}\left(f+g\right){\mathbf{=}}{\mathbf{T}}\left(f\right){\mathbf{+}}{\mathbf{T}}\left(g\right)\phantom{\rule{0ex}{0ex}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{T}}\left(\mathrm{kf}\right){\mathbf{=}}{\mathbf{kT}}\left(f\right)$

For all elements ${\mathbf{f}}{\mathbf{,}}{\mathbf{g}}$ of V and k is scalar.

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

## Step2: Explanation of the solution

Consider the transformation as follows.

$\mathrm{T}:{\mathrm{P}}_{3}\to {\mathrm{R}}^{3}$

Since,role="math" localid="1659418414775" $\mathrm{dim}\left({\mathrm{P}}_{3}\right)=4$ and $\mathrm{dim}\left({\mathrm{R}}^{3}\right)=3$

Therefore, $\mathrm{dim}\left({\mathrm{P}}_{3}\right)\ne \mathrm{dm}\left({\mathrm{R}}^{3}\right)$.

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 ${\mathrm{P}}_{3}$ to ${\mathrm{R}}^{3}$.