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

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 :)