Here, we show that the Matrix multiplication function representation of any linear function is unique. First, let us recall what is a linear function. A function is linear f : R" > R if 1. flu+v)= f(u)+ f(v) for all u,v R. 2. f(au) = af(u) for all u R and a R. Now, show that the function f(z) = Az where A R*" is linear. Also show that if x # 0 and Az = Bz, then, A = B. Next, you need to show that any linear function f can be represented by f(x) = Az. In order to show that, you have to show a few things. 1. First, show that any vector x = {xy,xs,...,z,} can be represented as a linear combination of the basis vector, i.e., z = Y |, x;e; where ; is a n-dimensional vector with 1 at the i-th element and the rest are zero. 2. Since f is linear, you have f(z) = f(3.._, ie;) = Y., f(eix;), using the above you should construct the A matrix such that f(z) = Az. 3. In order to show uniqueness, you can use proof by contradiction. In par- ticular, assume that v # w, and yet f(v) = f(w), then show that you must have v = w contradicting your assumption