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

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 $E \subseteq X$ be a non-empty set.

Let $\lambda \in K$.

Then:

$\map T {\lambda E} = \lambda \map T E$

where $\lambda E$ denotes the dilation of $E$ by $\lambda$.

Proof

We have:

$y \in \map T {\lambda E}$

if and only if:

$y = \map T {\lambda x}$ for some $x \in E$.

From the linearity of $T$, this is equivalent to:

$y = \lambda T x$

This is equivalent to:

$y \in \lambda \map T E$

So by the definition of set equality we have:

$\map T {\lambda E} = \lambda \map T E$

$\blacksquare$