Consider two linear transformations y = T (x) and z = L (y), where T goes from R^m to R^p and L goes from R^p to R^n. Is the transformationz = L (T (x)) linear as well? [The transformation z = L (T (x)) is called the composite of T and L.]

Yes.

Proof: Consider x, y in R^m. Then since T is linear, we have:

T (a*x + b*y) = a*T (x) + b*T (y)

But since L is linear, we have:

L (a*T (x) + b*T (y)) = a*L (T (x)) + b*L (T (y))

So:

L (T (a*x + b*y)) = a*L (T (x)) + b*L (T (y))

and the composition is linear.