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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.

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2 Answers 2

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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.

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  • $\begingroup$ Very enlightening. And to think it was this simple... thanks! $\endgroup$
    – Sigma
    May 17, 2019 at 19:35
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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.

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