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<article class="md-content__inner md-typeset">
<h1 id="_1">向量空间<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>向量空间构成了机器学习的数学舞台。本文涵盖向量加法、标量乘法、封闭性公理、子空间，以及为什么AI中几乎所有东西都表示为向量。</em></p>
<ul>
<li>
<p>将向量空间想象成一种特定类型的舞台，数学对象生活在其中，每个对象被称为一个<strong>向量</strong>。</p>
</li>
<li>
<p>为了机器学习（ML）中的几何直觉，我们始终将向量视为欧几里得空间中的一个点，由其坐标表示。</p>
</li>
<li>
<p>向量 <span class="arithmatex">\(\mathbf{a}\)</span>（数学上用粗体小写字母表示）有 <span class="arithmatex">\(n\)</span> 个坐标，每个坐标代表沿一个轴的位置。</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{a} = [a_1, a_2, a_3]\]</div>
<p><img alt="向量 a = (3, 2, 4) 在三维空间中沿 x、y、z 轴绘制" src="../../images/vector_3d.svg" /></p>
<ul>
<li>
<p>向量空间中的向量遵循一套非常具体、不可打破的规则：</p>
<ul>
<li>
<p><strong>向量加法（组合）</strong>：
你可以取任意两个向量并将它们组合起来创建新向量。
把向量想象成移动的指令。
如果向量 A 表示"向前走 3 步"，向量 B 表示"向右走 2 步"，
将它们相加（A + B）就创建了一条新的单一指令："向前走 3 步并向右走 2 步。"</p>
</li>
<li>
<p><strong>标量乘法（缩放）</strong>：
你可以使用一个普通数字（"标量"）来缩放任意向量。
你可以拉伸它、缩小它或反转它。
如果向量 A 是"向前走 3 步"，将其乘以 2 就变成"向前走 6 步。"
将其乘以 -1 则完全翻转成"向后走 3 步。"</p>
</li>
</ul>
</li>
<li>
<p>向量空间的<strong>维度</strong>是其包含的独立方向的数量。<span class="arithmatex">\(\mathbb{R}^2\)</span> 是二维的（需要 2 个坐标），而上面的 <span class="arithmatex">\(\mathbf{a}\)</span> 存在于 <span class="arithmatex">\(\mathbb{R}^3\)</span> 中。</p>
</li>
<li>
<p>例如，我们可以将任何对象（比如一个人）表示为一个向量，其中 <span class="arithmatex">\(h_1\)</span> = 身高（厘米），<span class="arithmatex">\(h_2\)</span> = 体重（公斤），<span class="arithmatex">\(h_3\)</span> = 年龄。</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{h} = [185, 75, 30]\]</div>
<ul>
<li>
<p>我们现在已经创建了一个包含表示人的向量的向量空间。</p>
</li>
<li>
<p>我们可以表示多个人，并观察他们之间的远近！</p>
</li>
</ul>
<p><img alt="将三个人表示为向量：Alice 和 Carol 很近，Alice 和 Bob 很远" src="../../images/human_vectors.svg" /></p>
<ul>
<li>
<p>我们可以添加更多特征，创建丰富的人体表示，在 ML 中通常称为特征向量。</p>
</li>
<li>
<p>你拥有的独特且有意义的特征越多，特征向量的描述性就越强，这是需要记住的一个重要因素。</p>
</li>
<li>
<p>超过 3 维后，向量变得非常难以直观检查，这催生了一个名为<strong>线性代数</strong>的数学领域。</p>
</li>
<li>
<p>现在，<strong>线性代数</strong>是研究向量、向量空间以及向量之间映射关系的学科。</p>
</li>
<li>
<p>我们在 AI/ML 中将几乎所有东西都表示为向量，这使得线性代数成为该领域的基石。</p>
</li>
<li>
<p>向量加法可以通过将一个向量放在另一个向量的尾部，然后从原点画到终点的可视化方式执行。</p>
</li>
</ul>
<p><img alt="向量加法：a（红色）加 b（蓝色）得到结果 a + b（绿色虚线）" src="../../images/vector_addition.svg" /></p>
<ul>
<li>
<p>对于两个向量 <span class="arithmatex">\(\mathbf{a} = (a_1, a_2)\)</span> 和 <span class="arithmatex">\(\mathbf{b} = (b_1, b_2)\)</span>：<span class="arithmatex">\(\mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2)\)</span></p>
</li>
<li>
<p>向量也可以相减，所有加法规则同样适用。</p>
</li>
<li>
<p>将向量乘以标量会在相同方向上按该因子缩放向量。</p>
</li>
</ul>
<p><img alt="标量乘法：v（红色）、2v（蓝色，加倍）、-v（紫色，反向）" src="../../images/scalar_multiplication.svg" /></p>
<ul>
<li>
<p>对于标量 <span class="arithmatex">\(c\)</span> 和向量 <span class="arithmatex">\(\mathbf{v} = (v_1, v_2)\)</span>：<span class="arithmatex">\(c\mathbf{v} = (cv_1, cv_2)\)</span></p>
</li>
<li>
<p><strong>加法封闭性</strong>：如果将向量空间中的任意两个向量相加，结果也属于同一空间：如果 <span class="arithmatex">\(\mathbf{u} \in V\)</span> 且 <span class="arithmatex">\(\mathbf{v} \in V\)</span>，则 <span class="arithmatex">\(\mathbf{u} + \mathbf{v} \in V\)</span></p>
</li>
<li>
<p><strong>标量乘法封闭性</strong>：如果将向量空间中的任意向量乘以标量，结果也属于同一空间：如果 <span class="arithmatex">\(\mathbf{v} \in V\)</span> 且 <span class="arithmatex">\(c \in F\)</span>，则 <span class="arithmatex">\(c\mathbf{v} \in V\)</span></p>
</li>
<li>
<p><strong>加法结合律</strong>：对于任意三个向量 <span class="arithmatex">\(\mathbf{u}\)</span>、<span class="arithmatex">\(\mathbf{v}\)</span> 和 <span class="arithmatex">\(\mathbf{w}\)</span>：<span class="arithmatex">\((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\)</span></p>
</li>
<li>
<p><strong>加法交换律</strong>：对于任意两个向量 <span class="arithmatex">\(\mathbf{u}\)</span> 和 <span class="arithmatex">\(\mathbf{v}\)</span>：<span class="arithmatex">\(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\)</span></p>
</li>
</ul>
<p><img alt="平行四边形法则：两条路径（先 u 后 v，或先 v 后 u）到达同一点" src="../../images/commutativity.svg" /></p>
<ul>
<li>
<p>通过平行四边形的两条路径都到达同一点。</p>
</li>
<li>
<p><strong>（零向量）</strong>：存在一个向量 <span class="arithmatex">\(\mathbf{0}\)</span>，使得对于任何向量 <span class="arithmatex">\(\mathbf{v}\)</span>：<span class="arithmatex">\(\mathbf{v} + \mathbf{0} = \mathbf{v}\)</span></p>
</li>
</ul>
<p><img alt="零向量：v + 0 = v" src="../../images/zero_vector.svg" /></p>
<ul>
<li><strong>加法逆元</strong>：对于每个向量 <span class="arithmatex">\(\mathbf{v}\)</span>，存在一个向量 <span class="arithmatex">\(-\mathbf{v}\)</span>，使得：<span class="arithmatex">\(\mathbf{v} + (-\mathbf{v}) = \mathbf{0}\)</span></li>
</ul>
<p><img alt="加法逆元：v（红色）和 -v（蓝色）抵消为零" src="../../images/additive_inverse.svg" /></p>
<ul>
<li><strong>分配律 1</strong>：对于任意标量 <span class="arithmatex">\(c\)</span> 和向量 <span class="arithmatex">\(\mathbf{u}\)</span>、<span class="arithmatex">\(\mathbf{v}\)</span>：<span class="arithmatex">\(c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}\)</span></li>
</ul>
<p><img alt="分配律：缩放和（金色）等于缩放后的向量之和" src="../../images/distributivity.svg" /></p>
<ul>
<li>
<p>缩放和（金色）与分别缩放向量再求和的结果相同。</p>
</li>
<li>
<p><strong>分配律 2</strong>：对于任意标量 <span class="arithmatex">\(c\)</span>、<span class="arithmatex">\(d\)</span> 和向量 <span class="arithmatex">\(\mathbf{v}\)</span>：<span class="arithmatex">\((c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}\)</span></p>
</li>
<li>
<p><strong>结合律</strong>：对于任意标量 <span class="arithmatex">\(c\)</span>、<span class="arithmatex">\(d\)</span> 和向量 <span class="arithmatex">\(\mathbf{v}\)</span>：<span class="arithmatex">\((cd)\mathbf{v} = c(d\mathbf{v})\)</span></p>
</li>
<li>
<p><strong>单位元</strong>：对于任何向量 <span class="arithmatex">\(\mathbf{v}\)</span>：<span class="arithmatex">\(1\mathbf{v} = \mathbf{v}\)</span>，其中 <span class="arithmatex">\(1\)</span> 是标量域中的乘法单位元。</p>
</li>
<li>
<p><strong>子空间</strong>就是大空间内部的一个较小舞台。把三维空间想象成一个房间。一张穿过房间中心的平坦纸片就是一个子空间，穿过中心的一根直导线也是子空间。</p>
</li>
<li>
<p>关键要求是子空间必须经过原点。如果你把那片纸移开中心，它就不再是子空间了，因为零向量不再位于其上。</p>
</li>
</ul>
<p><img alt="子空间：经过原点的直线和平面在三维空间内部" src="../../images/subspaces.svg" /></p>
<ul>
<li>
<p>向量空间的所有规则（加法、缩放、封闭性）在子空间内部仍然有效。你可以在子空间内添加或缩放向量，永远不会"掉出"到更大的空间。</p>
</li>
<li>
<p>经过原点的直线是一维子空间，经过原点的平面是二维子空间，而整个空间是自身的子空间。</p>
</li>
<li>
<p>在 ML 中，子空间自然出现。高维数据通常具有存在于低维子空间上的结构。PCA 等技术找到那个子空间，这样我们可以更高效地处理数据。</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>运行代码验证分配律性质，然后修改并尝试测试其他规则！
<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">u</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="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></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="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">c</span> <span class="o">=</span> <span class="mi">2</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">lhs</span> <span class="o">=</span> <span class="n">c</span> <span class="o">*</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="n">v</span><span class="p">)</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">rhs</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">u</span> <span class="o">+</span> <span class="n">c</span><span class="o">*</span><span class="n">v</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;LHS: </span><span class="si">{</span><span class="n">lhs</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;RHS: </span><span class="si">{</span><span class="n">rhs</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>运行代码可视化不同的向量，然后修改不同坐标的值以理解每个轴如何影响位置。
<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><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="c1"># 尝试修改这些向量！</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">a</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="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">b</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="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">c</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="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s2">&quot;3d&quot;</span><span class="p">)</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="k">for</span> <span class="n">vec</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">a</span><span class="p">,</span> <span class="s2">&quot;a&quot;</span><span class="p">,</span> <span class="s2">&quot;red&quot;</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="s2">&quot;blue&quot;</span><span class="p">),</span> <span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="s2">&quot;c&quot;</span><span class="p">,</span> <span class="s2">&quot;green&quot;</span><span class="p">)]:</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a> <span class="n">ax</span><span class="o">.</span><span class="n">quiver</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">*</span><span class="n">vec</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">arrow_length_ratio</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">name</span><span class="p">)</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">lim</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">]))</span><span class="o">.</span><span class="n">max</span><span class="p">())</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="n">lim</span><span class="p">]);</span> <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="n">lim</span><span class="p">]);</span> <span class="n">ax</span><span class="o">.</span><span class="n">set_zlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="n">lim</span><span class="p">])</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;X&quot;</span><span class="p">);</span> <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Y&quot;</span><span class="p">);</span> <span class="n">ax</span><span class="o">.</span><span class="n">set_zlabel</span><span class="p">(</span><span class="s2">&quot;Z&quot;</span><span class="p">)</span>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></p>
</li>
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