Is Dimension 3 Enough to Prove Isomorphism Between Spaces?

[Daaavde]

Is it correct to state that if a space $E$ has dimension 3 then:
$E = ℝ^{3}$ and that the two spaces are isomorph?

 

[jgens]

If E is a 3-dimensional real vector space, then yes E is isomorphic as a vector space to R³. The statement that E = R³ is false however.

 

[Daaavde]

Then I wonder why in my textbook, every time there is an omomorphism $f$ whose image $Im(f)$ (row space) has dimension 3 it writes $Im(f) = ℝ^3$.

Am I missing something?

 

[jgens]

Two possibilities for this: First if your linear transformation has codomain R³ and its image has dimension 3, then its image literally is R³. And two sometimes the equality sign is used to mean "isomorphic to" and in that case I am f is definitely isomorphic to R³.

 

[Daaavde]

I think I should have add that all the homomorphisms in the textbook are always $f : ℝ^m \rightarrow ℝ^n$

 

[jgens]

In that case it is a slight abuse of notation to write I am f = R³, but also a very common one.

 

[Daaavde]

I see, thank you for your answer.