Isomorphism of Vector Spaces

Let $V$ and $W$ be two vector spaces over $\mathbb R$ and there is a bijection $T:V\rightarrow W$ which preserves addition, that is, for all vectors $u$ and $v$ in $V$ ,

$T(u+v)=T(u)+T(v).$

Must $V$ and $W$ be isomorphic?

If yes, prove the statement. Otherwise, give a counterexample.

#Algebra

Easy Math Editor

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Comments

Using the axiom of choice, we can see that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as $\mathbb{Q}$ -vector spaces, but obviously they are not isomorphic as $\mathbb{R}$ -vector spaces.

That is, here's a counterexample: set $V = \mathbb{R}, W = \mathbb{R}^2$ and set $T : V\to W$ equal to some isomorphism of $\mathbb{Q}$ -vector spaces.

Brian Moehring - 2 years, 8 months ago

Could you please tell me how to construct the isomorphism of $\mathbb Q$ -vector spaces from $\mathbb R$ onto $\mathbb R^2$ in detail?

Brian Lie - 2 years, 8 months ago

As I mentioned, it requires the axiom of choice, so it's not constructive.

The proof that such an isomorphism exists amounts to showing the following:

If $V$ is an uncountable $\mathbb{Q}$ -vector space, then $\dim_\mathbb{Q}(V) = |V|$
$\mathfrak{c} = \mathfrak{c}^2$

Together, this gives the existence of a bijection between the $\mathbb{Q}$ -bases of $\mathbb{R}$ and $\mathbb{R}^2,$ which can be extended linearly to a vector space isomorphism.

Brian Moehring - 2 years, 8 months ago

@Brian Moehring – Thanks.

Brian Lie - 2 years, 8 months ago

Here's the version in module:

Must group isomorphism between two $\mathbb R$ -modules be module isomorphism?

Brian Lie - 2 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$