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.

Horizontal change or the change of x