[gpm]

By the way, the definition of a vector space means we have a 0 vector, and the distribute property means that multiplication has to respect it.

v * w = v * (w + 0) # Vector space property that w + 0 = w

= v * w + v * 0 # Distributive property

Another vector space property guarantees that there exists -(v * w) such that -(v * w) + (v * w) = 0. Add it to both sides

-(v * w) + v * w = -(v * w) + v * w + v * 0

0 = 0 + v * 0 # And then reduce based on the definition of -(v * w)

0 = v * 0 # Vector space property that 0 + x = x

So you have your icing :)