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http:///mymemo.xyz/wiki/api.php?action=feedcontributions&user=114.49.9.126&feedformat=atom 2024-05-04T13:43:14Z 利用者の投稿記録 MediaWiki 1.31.0 2017-02-25T02:30:47Z

<p>114.49.9.126: 125.202.196.14 (会話) による ID:63140345 の版を取り消し</p> <hr /> <div>[[数学]]、とくに[[線型代数学]]において、&#039;&#039;&#039;線型部分空間&#039;&#039;&#039;（せんけいぶぶんくうかん、linear subspace）または&#039;&#039;&#039;部分ベクトル空間&#039;&#039;&#039;（ぶぶんベクトルくうかん、vector subspace）とは、[[ベクトル空間]]の[[部分集合]]で、それ自身が元の空間の演算により線型空間になっているもののことである。<br /> <br /> ベクトル空間のある部分集合が、それ自身ある演算に関してベクトル空間の構造を持っていたとしても、その演算がもとの空間の演算でないならば部分線型空間とは呼ばない、ということに注意されたい。また、文脈により紛れの恐れのない場合には、線型部分空間のことを単に&#039;&#039;&#039;部分空間&#039;&#039;&#039;と呼ぶことがある。<br /> <br /> ==定義==<br /> [[可換体|体]] &#039;&#039;K&#039;&#039; 上のベクトル空間 &#039;&#039;L&#039;&#039; の空でない<br /> 部分集合 &#039;&#039;S&#039;&#039; &amp;sube; &#039;&#039;L&#039;&#039; に対して、和やスカラー積は元の線型空間 &#039;&#039;L&#039;&#039; で定義された演算として、<br /> # &#039;&#039;a&#039;&#039; + &#039;&#039;b&#039;&#039; &amp;isin; &#039;&#039;S&#039;&#039;<br /> # α&#039;&#039;a&#039;&#039; &amp;isin; &#039;&#039;S&#039;&#039;<br /> (for all &#039;&#039;a&#039;&#039;, &#039;&#039;b&#039;&#039; &amp;isin; &#039;&#039;S&#039;&#039; and for all α &amp;isin; &#039;&#039;K&#039;&#039;) が満たされるとき、&#039;&#039;S&#039;&#039; を &#039;&#039;L&#039;&#039; の&#039;&#039;&#039;線型部分空間&#039;&#039;&#039;と呼ぶ。<br /> <br /> == 例 ==<br /> * ベクトル空間 &#039;&#039;V&#039;&#039; 自身や &#039;&#039;V&#039;&#039; の零元だけから成る集合 {0} は &#039;&#039;V&#039;&#039; の部分空間である。これを&#039;&#039;&#039;自明な部分空間&#039;&#039;&#039;という。<br /> * &#039;&#039;K&#039;&#039; 上のベクトル空間 &#039;&#039;V&#039;&#039; の任意の元 &#039;&#039;v&#039;&#039; に対して、集合 &#039;&#039;Kv&#039;&#039; = {&#039;&#039;av&#039;&#039; | &#039;&#039;a&#039;&#039; &amp;isin; &#039;&#039;K&#039;&#039;} は &#039;&#039;V&#039;&#039; の線型部分空間である。これを &#039;&#039;v&#039;&#039; の&#039;&#039;&#039;生成する線型部分空間&#039;&#039;&#039;という。<br /> <br /> * &#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt; や &#039;&#039;&#039;C&#039;&#039;&#039;&lt;sup&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt; に対し、&#039;&#039;原点を含む&#039;&#039; [[直線]]、[[平面]]、[[超平面]]は、全て線型部分空間である。<br /> *: &#039;&#039;&#039;注意&#039;&#039;&#039;：&#039;&#039;原点を含まない&#039;&#039; 直線、平面、超平面は線型部分空間とはならないが、これらは線型部分空間の概念と深く結びついている。実際、これらの概念を定義するときには、線型部分空間の概念を使うのが普通である（[[ユークリッド幾何学]]の&#039;&#039;古典的な&#039;&#039;公理系では、これらの用語は[[無定義語]]となる）。正確にはこれらは、アフィン部分空間とよばれるものである。詳しくは[[アフィン空間]]の項を参照。<br /> <br /> == 性質 ==<br /> ベクトル空間 &#039;&#039;V&#039;&#039; の線型部分空間 &#039;&#039;U&#039;&#039;, &#039;&#039;W&#039;&#039; に対し、その和<br /> : &#039;&#039;U&#039;&#039; + &#039;&#039;W&#039;&#039; = {&#039;&#039;u&#039;&#039; + &#039;&#039;w&#039;&#039; | &#039;&#039;u&#039;&#039; &amp;isin; &#039;&#039;U&#039;&#039;, &#039;&#039;w&#039;&#039; &amp;isin; &#039;&#039;W&#039;&#039;}　<br /> と[[共通部分|交わり]]<br /> : &#039;&#039;U&#039;&#039; &amp;cap; &#039;&#039;W&#039;&#039; = {&#039;&#039;v&#039;&#039; | &#039;&#039;v&#039;&#039; &amp;isin; &#039;&#039;U&#039;&#039; かつ &#039;&#039;v&#039;&#039; &amp;isin; &#039;&#039;W&#039;&#039;} <br /> も &#039;&#039;V&#039;&#039; の線型部分空間である。<br /> <br /> また、&#039;&#039;V&#039; &#039;&#039; も &#039;&#039;K&#039;&#039; 上の線型空間であって &#039;&#039;f&#039;&#039; が &#039;&#039;V&#039;&#039; から &#039;&#039;V&#039; &#039;&#039; への[[線型写像]]であるとき、&#039;&#039;V&#039;&#039; の任意の線型部分空間 &#039;&#039;W&#039;&#039; に対して<br /> : &#039;&#039;f&#039;&#039;(&#039;&#039;W&#039;&#039;) = {&#039;&#039;f&#039;&#039;(&#039;&#039;w&#039;&#039;) | &#039;&#039;w&#039;&#039; &amp;isin; &#039;&#039;W&#039;&#039;} <br /> は &#039;&#039;V&#039;&#039; の線型部分空間であり、&#039;&#039;V&#039; &#039;&#039; の任意の線型部分空間 &#039;&#039;W&#039; &#039;&#039; に対して <br /> :&#039;&#039;f&#039;&#039;&lt;sup&gt; -1&lt;/sup&gt;(&#039;&#039;W&#039; &#039;&#039;) = {&#039;&#039;v&#039;&#039; &amp;isin; &#039;&#039;W&#039;&#039; | &#039;&#039;f&#039;&#039;(&#039;&#039;v&#039;&#039;) &amp;isin; &#039;&#039;W&#039; &#039;&#039;} <br /> は &#039;&#039;V&#039; &#039;&#039; の線型部分空間である。特に、&#039;&#039;f&#039;&#039; の像 Im &#039;&#039;f&#039;&#039; = &#039;&#039;f&#039;&#039;(&#039;&#039;V&#039;&#039;)、[[零空間|核]] Ker &#039;&#039;f&#039;&#039; = &#039;&#039;f&#039;&#039;&lt;sup&gt; -1&lt;/sup&gt;({0&#039;}) は、それぞれ &#039;&#039;V&#039; &#039;&#039;, &#039;&#039;V&#039;&#039; の線型部分空間である。ただし 0&#039; は &#039;&#039;V&#039; &#039;&#039; の零元を表す。<br /> <br /> == 関連項目 ==<br /> * [[ベクトル空間]]<br /> * [[部分空間]]<br /> * [[代数系]]<br /> * [[アフィン空間]]<br /> <br /> == 外部リンク ==<br /> * {{MathWorld|urlname=Subspace|title=Subspace}}<br /> * {{PlanetMath|urlname=VectorSubspace|title=vector subspace}}<br /> <br /> {{DEFAULTSORT:せんけいふふんくうかん}}<br /> [[Category:線型代数学]]<br /> [[Category:数学に関する記事]]<br /> <br /> [[ru:Векторное пространство#Подпространство]]</div>

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