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I'm reading Luenberger's Optimization text, and I'm confused by a definition:

Let $S$ be a subset of a vector space $X$. The linear variety generated by $S$, denoted $v(S)$, is the intersection of all linear varieties in $X$ that contain $S$. A linear variety is defined as a translation of a subspace.

Can someone give me a couple examples of a linear variety generated by a subset of a vector space? I'm having trouble visualizing this.

lithium123
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    I guess you mean 'subspace' as 'linear variety'. Or what? – Berci Jun 23 '19 at 21:11
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    I'm just quoting straight out of Luenberger. Not quite sure either. – lithium123 Jul 26 '19 at 18:10
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    But it's written: 'A linear variety is defined as a translation of a subspace'. If 'translation' is not used in the meaning, like from one language to another, then it probably means shift by some vector $v$, i.e. the map $x\mapsto x+v$, and then 'linear variety' means affine subspace in modern terminology. More context could help to clarify it. – Berci Jul 26 '19 at 18:19
  • A footnote in Luenberger states that alternative terms for linear variety include an affine subspace, but he uses linear variety throughout, where linear variety $V = x_0 + M$ where $M$ is a subspace. Like the original poster, I'm having trouble. My main question comes down to whether vectors in some vector space $X$ must begin at the origin of $X$. Given any $S \subset X$, must each $S$ begin at $\theta_X$, the origin of $X$? If so, how would any affine space $V$ in $X$ contain the elements, if containing them "requires" that they begin at $\theta_V$? – akm May 14 '20 at 23:40
  • I think this thread may help. – akm May 15 '20 at 00:24
  • @lithium123 Are you still having trouble with this? – user56202 Jan 28 '21 at 03:30
  • @akm Are you still having trouble with this? – user56202 Jan 28 '21 at 03:31

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