Composition of Linear Transformations is Linear Transformation

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:

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