Let $A$ be an [affine space](/page/Affine%20Space) modeled on a $k$-[vector space](/page/Vector%20Space) $V$, and let $B$ be an affine space modeled on a $k$-vector space $W$. Let
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\begin{align*}
f:A \to B
\end{align*}
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be an affine isomorphism with linear part
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\begin{align*}
L:V \to W.
\end{align*}
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Then $L$ is a vector-space isomorphism. If $C \subset A$ is a nonempty affine subspace, then $f(C) \subset B$ is a nonempty affine subspace, and