Showing 2 vector spaces are isomorphic. I am trying to understand how to show two vector spaces are isomorphic. You do this by showing there is an isomorphism that can be mapped between the two spaces.
What I don't understand is my lecturer's way of showing that the isomorphism is one-to-one? 

It seems to me that he only showed that f is one-to one for the case of $w=0$. This, doesn't generalize to other points does it? 

Let $V$ and $W$ be vector spaces and let $T:V\to W$ be a linear transformation. In general, $T$ is one-to-one if and only if 
$$\ker{T} = \{v\in V\ |\ Tv=0_W\} = \{0_V\},$$
where $0_V$ is the zero vector in $V$. The fact that you may be looking at is that $T$ is one-to-one if and only if $w\in\ker{T}$ implies $w=0_V$.
