Image of Linear Combination of Subsets of Vector Space under Linear Transformation

Theorem
Let $K$ be a field.

Let $X$ and $Y$ be vector spaces over $K$.

Let $T : X \to Y$ be a linear transformation.

Let $\lambda, \mu \in K$.

Let $A, B \subseteq X$.

Then:
 * $T \sqbrk {\lambda A + \mu B} = \lambda T \sqbrk A + \mu T \sqbrk B$

Proof
We have: