Composition of Linear Transformations is Linear Transformation

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Theorem

Let $K$ be a field.

Let $X, Y, Z$ be vector spaces over $K$.

Let $T_1 : X \to Y$ and $T_2 : Y \to Z$ be linear transformations.


Then the composition $T_2 \circ T_1 : X \to Z$ is a linear transformation.


Proof

Let $\lambda \in K$ and $u, v \in X$.

Then, we have:

\(\ds \map {\paren {T_2 \circ T_1} } {\lambda u + v}\) \(=\) \(\ds \map {T_2} {\map {T_1} {\lambda u + v} }\)
\(\ds \) \(=\) \(\ds \map {T_2} {\lambda T_1 u + T_1 v}\) Definition of Linear Transformation
\(\ds \) \(=\) \(\ds \lambda \paren {T_2 T_1} u + \paren {T_2 T_1} v\)
\(\ds \) \(=\) \(\ds \lambda \paren {T_2 \circ T_1} u + \paren {T_2 \circ T_1} v\)

so $T_2 \circ T_1 : X \to Z$ is a linear transformation.

$\blacksquare$