Okay, so the following actually isn't a theorem asserted by the book I'm using. It's actually an example given to illustrate the importance of isomorphisms. It was a specific example with specific numbers but I want to try and prove that it holds in general. So, here's the assertion:
Let $f:V \to W$ be an isomorphism. Let $U_1$ and $U_2$ be subspaces of V. Then, the following holds:
$$\dim(U_1 \cap U_2) = \dim(f(U_1) \cap f(U_2))$$
Proof Attempt:
Let $(v_1,v_2,….,v_n)$ be a basis for $U_1 \cap U_2$. Thus, that n-tuple is linearly independent and spans the intersection of the two subspaces.
By a previously proven result, if $(v_1,v_2,….,v_n)$ is a linearly independent list of vectors, then $\beta = (f(v_1),f(v_2),….,f(v_n))$ is a linearly independent list of vectors. To show that this list forms a basis for $f(U_1) \cap f(U_2)$, we just need to show that it generates that set.
Consider the following linear combination:
$\sum_{k=1}^{n} \alpha_k f(v_k)$
Due to the linearity of f, we can rewrite the above as follows:
$f[\sum_{k=1}^{n} \alpha_k v_k]$.
$L(\beta) \subset f(U_1) \cap f(U_2)$. So, we have to prove containment in the other direction. Let $w \in f(U_1) \cap f(U_2)$. Then, $w \in f(U_1)$ and $w \in f(U_2)$.
By the surjectivity of f, there exists a vector $u_1 \in U_1$ such that $f(u_1) = w$ and there exists a vector $u_2 \in U_2$ such that $f(u_2) = w$. By the injectivity of f, it's clear that $u_1 = u_2$.
That also means that $u_1 \in U_1 \cap U_2$ so it can be written as a linear combination of the vectors $(v_1,v_2,…,v_n)$. Therefore, $f(u_1)$ can be written as a linear combination of the vectors in $\beta$ and this proves that $f(U_1) \cap f(U_2) \subset L(\beta)$. This proves that $\beta$ generates $f(U_1) \cap f(U_2)$.
So, $\beta$ forms a basis for $f(U_1) \cap f(U_2)$ and that basis has length $n$. Similarly, the basis for $U_1 \cap U_2$ also has length n. Thus, we conclude that:
$\dim(U_1 \cap U_2) = \dim(f(U_1) \cap f(U_2))$.
Once again, I need to know if my argument above is correct or not. I'm also pretty sure that there's a short and slick proof of this but I certainly can't see it (maybe with the use of set identities?).
Best Answer
Yes, I think your proof is correct. And in general you don't need the basis to prove this.
Claim:
$$g:U_1\cap U_2\rightarrow f(U_1)\cap f(U_2)\\x\mapsto f(x)$$ is an isomorphism, if $f:V\rightarrow W$ is an injective linear map.
Proof:
Suppose $g(x)=f(x)=0$ then by the injectivity of $f$, $x=0$, so $g$ is injective.
Take any $y\in f(U_1)\cap f(U_2)$. By the definition of $f(U_1)$ and $f(U_2)$ there exist $x_1\in U_1$ and $x_2\in U_2$ such that $y=f(x_1)=f(x_2)$. Since $f$ is injective, we see $x_1=x_2\in U_1\cap U_2$. So $y=g(x_1)$. Thus $g$ is surjective.
Hence $g$ is a bijective linear transformation.
By set-theory, we know $g$ has a unique inverse function $g^{-1}:f(U_1)\cap f(U_2)\rightarrow U_1\cap U_2$. Then, for any $x,y\in f(U_1)\cap f(U_2)$ and $f\in F$, where $F$ is the base field, we have $$ g(g^{-1}(fx+y))=fx+y=fg(g^{-1}(x))+g(g^{-1}(y))=g(fg^{-1}(x)+g^{-1}(y)). $$ Since $g$ is injective, this shows that $g^{-1}$ is linear. Therefore $g$ is a linear isomorphism.
Hope this helps.