# Show that $S \circ T : V \longrightarrow W_1$ is a linear transformation

Let $$T : V \longrightarrow W$$ and $$S : W \longrightarrow W_1$$ be linear transformations. Show that $$S \circ T : V \longrightarrow W_1$$ is a linear transformation

Thoughts on this problem:

Usually, when problems ask you to show a transformation is linear, they give you the rule of the transformation. Here I'm asked to show that the composition of two generic linear transformations is also a linear transformation, and I don't know how to do that without the actual equations for $$S$$ and $$T$$. I don't know how to prove the statement at this level of generality.

And yet, the statement seems rather intuitive. For example, if you represent two linear transformations with matrices, then $$S \circ T$$ is just $$ST$$ and then you find $$ST$$ by multiplying the matrices. Easy stuff.

You don't need the rules, it is enough to know that $$S$$ and $$T$$ are linear. For any two vectors $$u,v$$ and any scalar $$\lambda$$:
$$S\circ T(u+v)=S(T(u+v))=S(T(u)+T(v))=S(T(u))+S(T(v))=S\circ T(u)+S\circ T(v)$$
$$S\circ T(\lambda u)=S(T(\lambda u))=S(\lambda T(u))=\lambda S(T(u))=\lambda S\circ T(u)$$
Hence $$S\circ T$$ is linear. That's it.
A linear transformation is a special kind of function between vector spaces. It has the properties of being additive and homogeneous. You are told that $$S,T$$ have these properties, and you must now verify them for $$S\circ T$$. No formulas for $$S,T$$ are needed.