Orientation of a Vector Space and the Homology of a pair

Question

I am stuck at the following argument in the book Charcteristic Classes by Milnor & Stasheff page 96:

A choice of orientation for V corresponds to a choice of one of the two possible generators for the singular homology group $H_n(V, V_0; Z)$.

I was able to proof that homology group above mentioned is actually isomorphic to Integers. This is what they have explained in the next few lines.

Choose some orientation preserving linear embedding $$\sigma : \Delta^n\to V $$ which maps the barycenter of $ \Delta^n$ to the zero vector,hence maps the boundary of $ \Delta^n$ into $V_0$.Then $ \Delta^n$ is a singular n-simplex representing an element in the group of relative n-cycles $Z_n(V, V_0; Z)$.

Here are my problems:

1. What about the existence of such a map.
2. If the barycentre is being mapped to zero then why it implies the boundary has to be non-zero?
3. Why does all this implies that it has to be a relative n-cycle.

Kindly help!! Any suggestions are valuable. Regards