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:

\(\ds T \sqbrk {S D}\)

\(=\)

\(\ds T \sqbrk {\bigcup_{s \mathop \in S} s D}\)

\(\ds \)

\(=\)

\(\ds \bigcup_{s \mathop \in S} T \sqbrk {s D}\)

Image of Union under Mapping

\(\ds \)

\(=\)

\(\ds \bigcup_{s \mathop \in S} s T \sqbrk D\)

Image of Dilation of Set under Linear Transformation is Dilation of Image

\(\ds \)

\(=\)

\(\ds S T \sqbrk D\)

$\blacksquare$