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<article class="md-content__inner md-typeset">
<h1 id="_1">基与对偶性<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>基定义了向量空间的坐标系，而对偶性揭示了线性函数如何作用于向量。本文涵盖线性无关性、生成集、基变换、对偶空间和余向量，这些概念支撑了 ML 中的 PCA、特征变换和注意力查询。</em></p>
<ul>
<li>
<p>我们已经看到向量存在于具有一定维度数的空间中。但什么定义了这些维度？这就是<strong>基向量</strong>发挥作用的地方。</p>
</li>
<li>
<p><strong>基</strong>是一组向量，可以通过缩放和相加（线性组合）构建空间中的每个其他向量，且没有冗余。它们是空间的构建块。</p>
</li>
<li>
<p>基必须满足两个条件：</p>
<ul>
<li>
<p><strong>线性无关</strong>：没有基向量能从其他基向量构造出来。每个都贡献了一个真正的新方向。</p>
</li>
<li>
<p><strong>生成性</strong>：空间中的每个向量都可以表示为基向量的组合。没有任何遗漏。</p>
</li>
</ul>
</li>
<li>
<p>基中的向量数量等于空间的<strong>维度</strong>。在 <span class="arithmatex">\(\mathbb{R}^2\)</span> 中你需要 2 个，在 <span class="arithmatex">\(\mathbb{R}^3\)</span> 中你需要 3 个，依此类推。</p>
</li>
<li>
<p>最自然的基是<strong>标准基</strong>，即沿每个轴的单位向量：</p>
<ul>
<li>在 <span class="arithmatex">\(\mathbb{R}^2\)</span> 中：<span class="arithmatex">\(\hat{\mathbf{i}} = (1, 0)\)</span> 和 <span class="arithmatex">\(\hat{\mathbf{j}} = (0, 1)\)</span></li>
<li>在 <span class="arithmatex">\(\mathbb{R}^3\)</span> 中：<span class="arithmatex">\(\hat{\mathbf{i}} = (1, 0, 0)\)</span>，<span class="arithmatex">\(\hat{\mathbf{j}} = (0, 1, 0)\)</span>，<span class="arithmatex">\(\hat{\mathbf{k}} = (0, 0, 1)\)</span></li>
</ul>
</li>
<li>
<p>任何向量都是这些基向量的加权和。向量 <span class="arithmatex">\((3, 2)\)</span> 实际上是 <span class="arithmatex">\(3\hat{\mathbf{i}} + 2\hat{\mathbf{j}}\)</span>。权重（3 和 2）是该基下向量的<strong>坐标</strong>。</p>
</li>
<li>
<p>但标准基并不是唯一有效的基。在 <span class="arithmatex">\(\mathbb{R}^2\)</span> 中，向量 <span class="arithmatex">\((1, 1)\)</span> 和 <span class="arithmatex">\((-1, 1)\)</span> 也构成基。它们线性无关，并且可以到达平面上的任何点。同一个向量在这个新基下只是有不同的坐标。</p>
</li>
<li>
<p><strong>基变换</strong>使用不同的构建块重新表达同一个向量。向量没有移动，我们只是从不同的角度描述它。</p>
</li>
<li>
<p>这是通过乘以一个<strong>基变换矩阵</strong> <span class="arithmatex">\(P\)</span> 来完成的，其列是用旧坐标表示的新基向量。要变回去，乘以 <span class="arithmatex">\(P^{-1}\)</span>。</p>
</li>
<li>
<p>在 ML 中，基变换经常出现。例如，PCA 找到一个新基（主成分），在该基下数据更容易理解，坐标轴与最大变化方向对齐。</p>
</li>
<li>
<p>现在，这里隐藏着一个更深层的想法。当我们写 <span class="arithmatex">\(\mathbf{v} = (3, 2)\)</span> 时，坐标 3 和 2 实际上是沿着每个基方向"测量" <span class="arithmatex">\(\mathbf{v}\)</span> 的结果。第一个坐标问"<span class="arithmatex">\(\hat{\mathbf{i}}\)</span> 在 <span class="arithmatex">\(\mathbf{v}\)</span> 中有多少？"，第二个问"<span class="arithmatex">\(\hat{\mathbf{j}}\)</span> 呢？"</p>
</li>
<li>
<p>这些测量中的每一个都是一个<strong>线性泛函</strong>，一个接受向量并返回单个标量的函数。所有这样的线性泛函的集合构成了<strong>对偶空间</strong> <span class="arithmatex">\(V^\ast\)</span>。</p>
</li>
<li>
<p>这样想：向量是被测对象，线性泛函是测量它们的<em>标尺</em>。对偶空间是所有可能的标尺的集合。</p>
</li>
<li>
<p>对于每个基 <span class="arithmatex">\(\{\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n\}\)</span>，存在一个对应的<strong>对偶基</strong> <span class="arithmatex">\(\{\mathbf{e}_1^\ast, \mathbf{e}_2^\ast, \ldots, \mathbf{e}_n^\ast\}\)</span>。每个对偶基向量恰好提取一个坐标：</p>
</li>
</ul>
<div class="arithmatex">\[
\mathbf{e}_i^\ast(\mathbf{e}_j) = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}
\]</div>
<ul>
<li>
<p><span class="arithmatex">\(\mathbf{e}_1^\ast\)</span> 作用于 <span class="arithmatex">\(\mathbf{e}_1\)</span> 时返回 1，对其它所有向量返回 0。它完美地隔离了第一个坐标。</p>
</li>
<li>
<p>点积连接了这两个世界。当你计算 <span class="arithmatex">\(\mathbf{u} \cdot \mathbf{v}\)</span> 时，你可以把其中一个向量看作"标尺"在测量另一个向量。点积 <span class="arithmatex">\(\mathbf{u} \cdot \mathbf{v}\)</span> 等同于将由 <span class="arithmatex">\(\mathbf{u}\)</span> 定义的线性泛函应用于向量 <span class="arithmatex">\(\mathbf{v}\)</span>。</p>
</li>
<li>
<p>这意味着每个向量都隐含地定义了一个线性泛函，并且每个线性泛函都可以用一个向量表示。在有限维空间中，对偶空间本质上是原始空间的镜像。</p>
</li>
<li>
<p>对偶性现在可能看起来很抽象，但它支撑着许多实际的概念：坐标是对偶基的评估，点积是对偶配对，而神经网络中的注意力等变换通过让一组向量"查询"另一组向量来运作，这正是对偶性在起作用。</p>
</li>
</ul>
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">&para;</a></h2>
<ol>
<li>
<p>在两个不同的基中表达一个向量，并验证它们代表同一个点。尝试创建你自己的基，观察向量得到什么坐标。
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">v</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="c1"># 标准基：坐标就是分量本身</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Standard basis coords: </span><span class="si">{</span><span class="n">v</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="c1"># 新基：(1,1) 和 (-1,1)</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">P</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">],</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">]])</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">new_coords</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;New basis coords: </span><span class="si">{</span><span class="n">new_coords</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="c1"># 验证：从新坐标重建</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">reconstructed</span> <span class="o">=</span> <span class="n">new_coords</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">new_coords</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Reconstructed: </span><span class="si">{</span><span class="n">reconstructed</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>验证对偶基性质：每个对偶基向量恰好提取一个坐标，对其他向量返回零。
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="c1"># R3 中的标准基</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="n">e1</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">e2</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">e3</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">v</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">])</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="c1"># 每个点积提取一个坐标</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;e1 · v = </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">e1</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">)</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;e2 · v = </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">e2</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">)</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;e3 · v = </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">e3</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">)</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
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