1. Matrices and Operations

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.

2. I don't fully understand your question, or the way the question is given:

Don't you mean $f:\mathbb{R}^m\to \mathbb{R}^n$ is defined by $f(x)= Ax$ where A is a $m\times n$ matrix?

3. Yes, that looks about right, the question itself made me struggle also.

4. In that case it's quite trivial:

$f(ax+by) = A(ax+by) = aAx+ bAy = af(x)+bf(y)$

Matrix miltiplication is linear.

5. Oh, ok. So say instead of y = Ax it was y = (A^2)x and y = (A^3)x.

Would the next step in solving them be

f(ax+by)^2 and f(ax+by)^3 respectively?

6. No, it would still be linear. The square or cube of a matrix is still a matrix so this is still just "multiplication by a matrix".

7. Thanks guys