Image of Pointwise Scalar Multiplication of Subset of Scalars with Subset of Vectors 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 $S \subseteq K$ and $D \subseteq X$ be non-empty sets.

Then:
 * $T \sqbrk {S D} = S T \sqbrk D$

where:
 * $S D = \set {\lambda x : \lambda \in S, \, x \in D}$

Proof
We have: