Theorem

${latex.inlinea0 = 0\quad \forall a \in F} Note that this is means that any scalar times the 0 vector is equal to the 0 vector.

Proof

For any ${latex.inlinea \in F} we have that ${latex.inlinea0 = a(0 + 0) = a0 + a0}. We can add the additive inverse of a0 to both sides and get that ${latex.inline0 = a0}.