Linear mapping/Shape/Translation/Image/Exercise

Let ${}F\subseteq \mathbb {R} ^{n}$ be a "geometric shape“, for example, a circle or a rectangle in the plane. Let

$\theta _{w}\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{n},x\longmapsto x+w,$

be the translation with the vector ${}w$, and let ${}F'=\theta _{w}(F)$ denote the translated shape. Let

$\varphi \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}$

be a linear mapping. Show that the image ${}\varphi (F')$ arises from the image ${}\varphi (F)$ by a translation.

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