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.
$\blacksquare$