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Q38E

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Found in: Page 200

### Linear Algebra With Applications

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

# TRUE OR FALSE?38. There exists a subspace of ${{\mathbf{ℝ}}}^{\mathbf{3}\mathbf{×}\mathbf{4}}$that is isomorphic to${{\mathbit{P}}}_{{\mathbf{9}}}$.

The given statement is true.

See the step by step solution

## Step 1: Isomorphism of vector space

Two vector spaces V and W over the same field F are isomorphic if there is a bisection ${\mathbit{T}}{\mathbf{:}}{\mathbit{V}}{\mathbf{\to }}{\mathbit{W}}$ which preserves addition and scalar multiplication, that is, for all vectors u and v in V, and all scalars ${\mathbf{c}}{\mathbf{\in }}{\mathbf{F}}{\mathbf{,}}{\mathbf{T}}{\mathbf{\left(}}{\mathbf{u}}{\mathbf{+}}{\mathbf{v}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{T}}{\mathbf{\left(}}{\mathbf{u}}{\mathbf{\right)}}{\mathbf{+}}{\mathbf{T}}{\mathbf{\left(}}{\mathbf{v}}{\mathbf{\right)}}{\mathbf{}}{\mathbf{and}}{\mathbf{}}{\mathbf{T}}{\mathbf{\left(}}{\mathbf{cv}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{cT}}{\mathbf{\left(}}{\mathbf{v}}{\mathbf{\right)}}$ .The correspondence T is called an isomorphism of vector spaces.

When $\mathrm{T}:\mathrm{V}\to \mathrm{W}$ is an isomorphism,$\mathrm{T}:\mathrm{V}\text{'}\to \mathrm{W}$ then it’s emphasize that it is an isomorphism When V and W, are isomorphic, but the specific isomorphism is not named, we’ll just write $V~=W.$.

## Step 2: condition of a subspace to be isomorphic.

Space${\mathrm{ℝ}}^{3×4}$ is 12-dimensional and ${P}_{9}$ is 10-dimensional. Spaces of same dimension are isomorphic, so any 10-dimensional subspace of ${\mathrm{ℝ}}^{3×4}$is isomorphic to ${P}_{9}$.

Hence, the statement is true.