Range$(M◦L)$ is a subspace of Range$(M)$

Question

Define the following linear mappings:

$$L:R^n→R^m$$

$$M:R^m → R^P$$

Prove that Range $(M◦L)$ is a subspace of Range $(M)$.

What I have so far (not sure if correct):

Range $(M◦L)=R^p$ and

Range $(M)=R^P$

So then I'd have to show that $R^p$ is a subspace of $R^p$?

Any thoughts on where to go? Thanks!

Answer 1

Note that $\DeclareMathOperator{Range}{Range}$ $$ \Range(M\circ L) = \bigl\{M\bigl(L(\vec v)\bigr): \vec v\in\Bbb R^n\bigr\} \subset \{M(\vec x):\vec x\in\Bbb R^m\} =\Range(M) $$