|
|
| Line 1: |
Line 1: |
| | + | {{TEX|done}} |
| | ''linear manifold, affine subspace'' | | ''linear manifold, affine subspace'' |
| | | | |
| − | A subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595301.png" /> of a (linear) [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595302.png" /> that is a translate of a [[Linear subspace|linear subspace]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595303.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595304.png" />, that is, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595305.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595306.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595307.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595308.png" /> determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l0595309.png" /> uniquely, whereas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953010.png" /> is defined only modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953011.png" />: | + | A subset $M$ of a (linear) [[vector space]] $E$ that is a translate of a [[linear subspace]] $L$ of $E$, that is, a set $M$ of the form $x_0 + L$ for some $x_0$. The set $M$ determines $L$ uniquely, whereas $x_0$ is defined only modulo $L$: |
| − | | + | $$ |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953012.png" /></td> </tr></table>
| + | x_0 + L = x_1 + N |
| − | | + | $$ |
| − | if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953014.png" />. The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953015.png" /> is the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059530/l05953016.png" />. A linear variety corresponding to a subspace of codimension 1 is called a hyperplane. | + | if and only if $L = N$ and $x_1 - x_0 \in L$. The dimension of $M$ is the dimension of $L$. A linear variety corresponding to a subspace of [[codimension]] 1 is called a ''hyperplane''. |
Revision as of 19:54, 31 December 2014
linear manifold, affine subspace
A subset $M$ of a (linear) vector space $E$ that is a translate of a linear subspace $L$ of $E$, that is, a set $M$ of the form $x_0 + L$ for some $x_0$. The set $M$ determines $L$ uniquely, whereas $x_0$ is defined only modulo $L$:
$$
x_0 + L = x_1 + N
$$
if and only if $L = N$ and $x_1 - x_0 \in L$. The dimension of $M$ is the dimension of $L$. A linear variety corresponding to a subspace of codimension 1 is called a hyperplane.
How to Cite This Entry:
Linear variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_variety&oldid=36015
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article