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The dimension of an affine space is defined as the dimension of the vector space of its translations.

What does mean here $\text{the vector space of its translations}$?

We know that an affine space do not have fixed origin. In other word, in an affine space no vector has a fixed origin and no vector can be uniquely associated to a point. Rather, in an affine space there are concept of displacement vectors or translation vectors between two points in the space.

But I can not understand the line $\text{the vector space of its translations}$.

Someone please explain it if possible with examples.

MAS
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1 Answers1

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An affine space $S$ is a set of the form $S=x+V$ where $x$ is a fixed vector and $V$ is a linear subspace. The dimension of this space is defined as dimension of $V$. ($V$ is a translate of $S$ because $V=\{-x+s: s\in S\}$).

Example: the line $y=x+1$ is affine in $\mathbb R^{2}$ and is is a translate of the one dimensional space $y=x$.

Kavi Rama Murthy
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