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lakerswillwin committed Apr 24, 2024
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318 changes: 270 additions & 48 deletions courses(the second semester)/linear algebra/index.html
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</span>
</a>

<nav class="md-nav" aria-label="商">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#定义商空间vu" class="md-nav__link">
<span class="md-ellipsis">
定义商空间(V/U)
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#平行于u的两个仿射子集或相等或不相交则有以下等价描述" class="md-nav__link">
<span class="md-ellipsis">
平行于U的两个仿射子集或相等或不相交，则有以下等价描述
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义vu上的加法和标量乘法" class="md-nav__link">
<span class="md-ellipsis">
定义V/U上的加法和标量乘法
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义商映射" class="md-nav__link">
<span class="md-ellipsis">
定义商映射
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#商空间的维数" class="md-nav__link">
<span class="md-ellipsis">
商空间的维数
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义一个映射t" class="md-nav__link">
<span class="md-ellipsis">
定义一个映射\(T^·\)
</span>
</a>

</li>

</ul>
</nav>

</li>

</ul>
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</span>
</a>

<nav class="md-nav" aria-label="对偶空间与对偶映射">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#线性泛函v上的线性泛函是从v到f的线性映射" class="md-nav__link">
<span class="md-ellipsis">
线性泛函：V上的线性泛函是从V到F的线性映射
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#对偶空间" class="md-nav__link">
<span class="md-ellipsis">
对偶空间
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义对偶基" class="md-nav__link">
<span class="md-ellipsis">
定义对偶基
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#对偶基是对偶空间的基" class="md-nav__link">
<span class="md-ellipsis">
对偶基是对偶空间的基
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义对偶映射-t" class="md-nav__link">
<span class="md-ellipsis">
定义对偶映射 \(T^,\)
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#对偶映射的代数性质" class="md-nav__link">
<span class="md-ellipsis">
对偶映射的代数性质
</span>
</a>

</li>

</ul>
</nav>

</li>

</ul>
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</span>
</a>

<nav class="md-nav" aria-label="商">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#定义商空间vu" class="md-nav__link">
<span class="md-ellipsis">
定义商空间(V/U)
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#平行于u的两个仿射子集或相等或不相交则有以下等价描述" class="md-nav__link">
<span class="md-ellipsis">
平行于U的两个仿射子集或相等或不相交，则有以下等价描述
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义vu上的加法和标量乘法" class="md-nav__link">
<span class="md-ellipsis">
定义V/U上的加法和标量乘法
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义商映射" class="md-nav__link">
<span class="md-ellipsis">
定义商映射
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#商空间的维数" class="md-nav__link">
<span class="md-ellipsis">
商空间的维数
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义一个映射t" class="md-nav__link">
<span class="md-ellipsis">
定义一个映射\(T^·\)
</span>
</a>

</li>

</ul>
</nav>

</li>

</ul>
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</span>
</a>

<nav class="md-nav" aria-label="对偶空间与对偶映射">
<ul class="md-nav__list">

<li class="md-nav__item">
<a href="#线性泛函v上的线性泛函是从v到f的线性映射" class="md-nav__link">
<span class="md-ellipsis">
线性泛函：V上的线性泛函是从V到F的线性映射
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#对偶空间" class="md-nav__link">
<span class="md-ellipsis">
对偶空间
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义对偶基" class="md-nav__link">
<span class="md-ellipsis">
定义对偶基
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#对偶基是对偶空间的基" class="md-nav__link">
<span class="md-ellipsis">
对偶基是对偶空间的基
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#定义对偶映射-t" class="md-nav__link">
<span class="md-ellipsis">
定义对偶映射 \(T^,\)
</span>
</a>

</li>

<li class="md-nav__item">
<a href="#对偶映射的代数性质" class="md-nav__link">
<span class="md-ellipsis">
对偶映射的代数性质
</span>
</a>

</li>

</ul>
</nav>

</li>

</ul>
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</li>
</ul>
<h4 id="商">商<a class="headerlink" href="#商" title="Permanent link"></a></h4>
<ul>
<li>定义商空间(V/U)
设U是V的子空间，则商空间V/U是指所有平行于U的仿射子集的集合
V/U={v+U：v∈V}</li>
<li>平行于U的两个仿射子集或相等或不相交，则有以下等价描述</li>
<h5 id="定义商空间vu">定义商空间(V/U)<a class="headerlink" href="#定义商空间vu" title="Permanent link"></a></h5>
<p>设U是V的子空间，则商空间V/U是指所有平行于U的仿射子集的集合
V/U={v+U：v∈V}</p>
<h5 id="平行于u的两个仿射子集或相等或不相交则有以下等价描述">平行于U的两个仿射子集或相等或不相交，则有以下等价描述<a class="headerlink" href="#平行于u的两个仿射子集或相等或不相交则有以下等价描述" title="Permanent link"></a></h5>
<ol>
<li>v-w∈V</li>
<li>v+U=w+U</li>
<li>(v+U)∩(w+U)≠空集</li>
<li>定义V/U上的加法和标量乘法
(v+U)+(w+U)=(v+w)+U
<span class="arithmatex">\(\lambda\)</span>(v+U)=<span class="arithmatex">\(\lambda\)</span>v+U</li>
<li>商空间是向量空间</li>
<li>定义商映射
商映射<span class="arithmatex">\(\pi\)</span>,对任意v属于V，<span class="arithmatex">\(\pi\)</span>(v)=v+U</li>
<li>商空间的维数
设V是有限维的，U是V的子空间，则dimV/U=dimV-dimU</li>
<li>
<p>定义一个映射<span class="arithmatex">\(T^·\)</span>
<span class="arithmatex">\(T^·\)</span>(v+nullT)=Tv
</ol>
<h5 id="定义vu上的加法和标量乘法">定义V/U上的加法和标量乘法<a class="headerlink" href="#定义vu上的加法和标量乘法" title="Permanent link"></a></h5>
<p>(v+U)+(w+U)=(v+w)+U
<span class="arithmatex">\(\lambda\)</span>(v+U)=<span class="arithmatex">\(\lambda\)</span>v+U</p>
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">1</span></pre></div></td><td class="code"><div><pre><span></span><code>* 商空间是向量空间
</code></pre></div></td></tr></table></div>
<h5 id="定义商映射">定义商映射<a class="headerlink" href="#定义商映射" title="Permanent link"></a></h5>
<p>商映射<span class="arithmatex">\(\pi\)</span>,对任意v属于V，<span class="arithmatex">\(\pi\)</span>(v)=v+U</p>
<h5 id="商空间的维数">商空间的维数<a class="headerlink" href="#商空间的维数" title="Permanent link"></a></h5>
<p>设V是有限维的，U是V的子空间，则dimV/U=dimV-dimU</p>
<h5 id="定义一个映射t">定义一个映射<span class="arithmatex">\(T^·\)</span><a class="headerlink" href="#定义一个映射t" title="Permanent link"></a></h5>
<p><span class="arithmatex">\(T^·\)</span>(v+nullT)=Tv
性质：</p>
</li>
<li>
<p><span class="arithmatex">\(T^·\)</span>是线性映射</p>
</li>
<ol>
<li><span class="arithmatex">\(T^·\)</span>是线性映射</li>
<li><span class="arithmatex">\(T^·\)</span>是单的</li>
<li>range<span class="arithmatex">\(T^·\)</span>=rangeT</li>
<li>V/(nullT)同构于rangeT</li>
</ul>
</ol>
<h3 id="对偶">对偶<a class="headerlink" href="#对偶" title="Permanent link"></a></h3>
<h4 id="对偶空间与对偶映射">对偶空间与对偶映射<a class="headerlink" href="#对偶空间与对偶映射" title="Permanent link"></a></h4>
<ul>
<li>
<p>线性泛函：V上的线性泛函是从V到F的线性映射</p>
</li>
<li>
<p>对偶空间：V上所有线性泛函构成的向量空间 记作<span class="arithmatex">\(V^<code>$ 且dimV=dim$V^</code>\)</span>(前提条件是V是有限维的)</p>
</li>
<li>
<p>定义对偶基</p>
</li>
</ul>
<h5 id="线性泛函v上的线性泛函是从v到f的线性映射">线性泛函：V上的线性泛函是从V到F的线性映射<a class="headerlink" href="#线性泛函v上的线性泛函是从v到f的线性映射" title="Permanent link"></a></h5>
<h5 id="对偶空间">对偶空间<a class="headerlink" href="#对偶空间" title="Permanent link"></a></h5>
<p>V上所有线性泛函构成的向量空间 记作<span class="arithmatex">\(V^<code>$ 且dimV=dim$V^</code>\)</span>(前提条件是V是有限维的)</p>
<h5 id="定义对偶基">定义对偶基<a class="headerlink" href="#定义对偶基" title="Permanent link"></a></h5>
<p>设v1,v2.....vn是V的基，则v1,v2......vn的对偶基是<span class="arithmatex">\(V^`\)</span>中的元素组<span class="arithmatex">\(\phi\)</span>1.....<span class="arithmatex">\(\phi\)</span>n,其中每个<span class="arithmatex">\(\phi\)</span>j都是V上的线性泛函，
使得<span class="arithmatex">\(\phi\)</span>j(vk)=1(当k=j)，0(当k≠j)</p>
<ul>
<li>
<p>对偶基是对偶空间的基</p>
</li>
<li>
<p>定义对偶映射 <span class="arithmatex">\(T^,\)</span></p>
</li>
</ul>
<h5 id="对偶基是对偶空间的基">对偶基是对偶空间的基<a class="headerlink" href="#对偶基是对偶空间的基" title="Permanent link"></a></h5>
<h5 id="定义对偶映射-t">定义对偶映射 <span class="arithmatex">\(T^,\)</span><a class="headerlink" href="#定义对偶映射-t" title="Permanent link"></a></h5>
<p>对于一个映射T，其对偶映射<span class="arithmatex">\(T^,\)</span>,且对于<span class="arithmatex">\(\phi\)</span>∈<span class="arithmatex">\(W^`\)</span>,<span class="arithmatex">\(T^,\)</span><span class="arithmatex">\(\phi\)</span>=<span class="arithmatex">\(\phi\)</span>*T</p>
<ul>
<li>
<p>对偶映射的代数性质</p>
</li>
<li>
<p>对所有的S,T有<span class="arithmatex">\((S+T)^,\)</span>=<span class="arithmatex">\(S^,\)</span>+<span class="arithmatex">\(T^,\)</span></p>
</li>
<h5 id="对偶映射的代数性质">对偶映射的代数性质<a class="headerlink" href="#对偶映射的代数性质" title="Permanent link"></a></h5>
<ol>
<li>对所有的S,T有<span class="arithmatex">\((S+T)^,\)</span>=<span class="arithmatex">\(S^,\)</span>+<span class="arithmatex">\(T^,\)</span></li>
<li></li>
</ul>
</ol>
<script>
MathJax = {
tex: {
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