Demonstration of Distributive Property

It is fairly easy to demonstrate that the distributive property is true. Let's do the demonstration with 3D column matrices, although it will work with any dimension. Both sides of the equation represent a real number. So the job is to check that both sides represent the same real number.

Show that: a · (b + c) = a · b + a · c
Let a = (f, g, h)^T
Let b = (r, s, t)^T
Let c = (x, y, z)^T

a · (b + c) = a · ( r+x, s+y, t+z )^T
= f(r+x) + g(s+y) + h(t+z)
= fr + fx + gs + gy + ht + hz

a · b + a · c = (fr + gs + ht) + (fx + gy + hz)
= fr + fx + gs + gy + ht + hz

Both sides of the equation represent the same real number.

The same demonstration could be done with vectors of any dimension, so the distributive property is true for any dimension.