Does isomorphism always map a subspace to a subspace? Let $\phi$ be an isomorphism between two real vector spaces  $V_1$ and $V_2$.
Is it always true that every subspace of $V_1$ is mapped by $\phi$ into a subspace of $V_2$, and vice versa?
If so, am I correct to say that there is a bijection between the set of subspaces of $V_1$ and the set of subspaces of $V_2$?
 A: Yes. If $V_1$ and $V_2$ are isomorphic, then they have the same shape. The isomorphism $\phi:V_1\to V_2$ provides a means of translating $V_1$-stuff into $V_2$-stuff. In particular, $\phi$ induces a bijection between the set of subspaces of $V_1$ and the set of subspaces of $V_2$.
This can be proven from the following three facts: 


*

*If $T:V\to W$ is a linear map then $\text{im } T$ is a subspace of $W$.

*If $U$ is a subspace of $V$ then $T|_U:U\to W$ is a linear map, and $\text{im }T|_U = T(U)$.

*If $T$ is an isomorphism, then $T^{-1}:W\to V$ exists and is a linear map.

A: You can check by picking your favorite subspace test, and making sure that the criteria are fulfilled. For example, restricting $\phi$ to an arbitrary subspace of $V_1$, is $0$ in the image? Is the image closed under vector sums, and scalar multiples?. 
Just saying that subspaces get mapped to subspaces isn't enough to guarantee a bijection, but since you have an isomorphism, it will be a bijection. You would need to show that the map $\phi : \text{Subspaces of } V_1 \to \text{Subspaces of } V_2$ is one-to-one and onto, or equivalently, is invertible.
