I don't fully understand your question, or the way the question is given:
Don't you mean is defined by where A is a matrix?
Let A be a n x m matrix. Show that if the function y = f(x) defined for m x 1 matrices x by y = Ax satisfies the linearity property, then f(aw + bz) = af(w) + bf(z) for any real numbers a and b and any m x 1 matrices w and z.
This is what I did
Scalars - k1, k2
Vectors - u, v
Matrix - n x m matrix
f(aw + bz) = af(w) + bf(z)
A(k1u + k2v) = k1Au + k2Av
m x 1 matrices are the vectors u and v
multiplying scalars to vectors:
mk1 x k1 matrix
mk2 x k2 matrix
I'm not understanding how I am supposed to prove that this function is linear.