Image of Translation of Set under Linear Transformation is Translation 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 $x \in X$.

Then:

$\map T {E + x} = \map T E + T x$

where $E + x$ denotes the translation of $E$ by $x$.

Proof

We have:

$y \in \map T {E + x}$

if and only if:

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

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

$y = T u + T x$ for some $u \in E$.

This is equivalent to:

$y \in \map T E + T x$

So by the definition of set equality we have:

$\map T {E + x} = \map T E + T x$

$\blacksquare$