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<article class="md-content__inner md-typeset">
<h1 id="_1">线性变换<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>每个矩阵乘法都是一个线性变换——一个在保持线性性质的同时重塑、旋转或投影向量的函数。本文涵盖旋转、反射、缩放、剪切、投影、映射的核与像，以及神经网络层如何串联这些变换。</em></p>
<ul>
<li>
<p><strong>线性变换</strong>（或线性映射）是一个接收向量并产生另一个向量的函数，同时保持加法和缩放性质。如果 <span class="arithmatex">\(T\)</span> 是线性的，则：</p>
<ul>
<li><span class="arithmatex">\(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\)</span></li>
<li><span class="arithmatex">\(T(c\mathbf{u}) = cT(\mathbf{u})\)</span></li>
</ul>
</li>
<li>
<p>每个线性变换都可以表示为矩阵乘法。矩阵<em>就是</em>变换本身。当你用一个矩阵乘以一个向量时，就是在对它施加一个线性变换。</p>
</li>
<li>
<p>可以把一个 <span class="arithmatex">\(2 \times 2\)</span> 矩阵想象成一个机器：它接收二维向量，输出新的二维向量。矩阵的列告诉你标准基向量 <span class="arithmatex">\(\hat{\mathbf{i}}\)</span> 和 <span class="arithmatex">\(\hat{\mathbf{j}}\)</span> 经过变换后到了哪里。其余一切都由线性性质导出。</p>
</li>
</ul>
<p><img alt="矩阵的列显示了基向量落在何处" src="../../images/basis_transform.svg" /></p>
<ul>
<li>例如，如果</li>
</ul>
<div class="arithmatex">\[
A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}
\]</div>
<p>那么 <span class="arithmatex">\(\hat{\mathbf{i}} = [1, 0]^T\)</span> 落在 <span class="arithmatex">\([2, 1]^T\)</span>（第1列），<span class="arithmatex">\(\hat{\mathbf{j}} = [0, 1]^T\)</span> 落在 <span class="arithmatex">\([1, 2]^T\)</span>（第2列）。其他所有向量都是这两个向量的组合，因此其输出自动遵循。</p>
<ul>
<li>
<p>将两个矩阵相乘可以理解为依次施加两个变换。如果 <span class="arithmatex">\(B\)</span> 将向量从一个空间变换，然后 <span class="arithmatex">\(A\)</span> 变换结果，那么 <span class="arithmatex">\(AB\)</span> 按顺序完成这两个操作。在游戏引擎中，先旋转角色再向前移动，与先移动再旋转，结果完全不同——这就是矩阵乘法不满足交换律的原因。</p>
</li>
<li>
<p><strong>旋转</strong>将向量绕一定角度 <span class="arithmatex">\(\theta\)</span> 转动而不改变其长度。向量大小不变，只是指向新的方向。</p>
</li>
</ul>
<p><img alt="旋转保持长度不变但改变方向" src="../../images/rotation.svg" /></p>
<ul>
<li>二维中的旋转矩阵为：</li>
</ul>
<div class="arithmatex">\[
R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}
\]</div>
<ul>
<li>当 <span class="arithmatex">\(\theta = 90°\)</span> 时：</li>
</ul>
<div class="arithmatex">\[
R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}
\]</div>
<p>因此 <span class="arithmatex">\([1, 0]^T\)</span> 变成 <span class="arithmatex">\([0, 1]^T\)</span>。原来指向右侧的向量现在指向上方。旋转矩阵是正交的，且行列式始终为1。当你在手机上旋转照片时，就是对每个像素坐标应用这个矩阵。</p>
<ul>
<li>在三维中，每个坐标轴都有对应的旋转矩阵。机械臂的每个关节绕特定轴旋转，每个关节就是一个旋转矩阵。绕z轴旋转看起来像是嵌入三维的二维情况：</li>
</ul>
<div class="arithmatex">\[
R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}
\]</div>
<ul>
<li><strong>缩放</strong>沿每个坐标轴独立地拉伸或压缩向量：</li>
</ul>
<div class="arithmatex">\[
S(s_x, s_y) = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}
\]</div>
<p><img alt="缩放沿每个轴以不同因子拉伸" src="../../images/scaling.svg" /></p>
<ul>
<li>
<p><span class="arithmatex">\(S(2, 1.5)\)</span> 将x分量加倍，y分量乘以1.5。沿某轴缩放 <span class="arithmatex">\(-1\)</span> 会翻转该分量。对角矩阵总是缩放变换。当你将图片缩小到50%时，就是对每个像素坐标应用 <span class="arithmatex">\(S(0.5, 0.5)\)</span>。</p>
</li>
<li>
<p><strong>反射</strong>像镜子一样将向量翻转到某个轴或直线的另一侧。沿x轴的反射保持x分量不变，取反y分量：</p>
</li>
</ul>
<div class="arithmatex">\[
\text{Ref}_x = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
\]</div>
<p><img alt="沿x轴反射翻转y分量" src="../../images/reflection.svg" /></p>
<ul>
<li>例如，<span class="arithmatex">\([3, 2]^T\)</span> 变成 <span class="arithmatex">\([3, -2]^T\)</span>。当你的手机水平翻转自拍照使文字正确显示时，就是在应用反射矩阵。沿直线 <span class="arithmatex">\(y = x\)</span> 的反射交换两个分量：</li>
</ul>
<div class="arithmatex">\[
\text{Ref}_{y=x} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\]</div>
<ul>
<li>
<p>反射矩阵的行列式为 <span class="arithmatex">\(-1\)</span>，表明它们翻转了方向。</p>
</li>
<li>
<p>旋转和反射都是<strong>刚性变换</strong>：它们保持距离和角度不变。表示这些变换的矩阵是正交矩阵，这就是为什么正交矩阵的行列式总是 <span class="arithmatex">\(+1\)</span>（旋转）或 <span class="arithmatex">\(-1\)</span>（反射）。</p>
</li>
<li>
<p><strong>剪切</strong>沿一个坐标轴按另一坐标轴的比例倾斜向量。水平剪切因子 <span class="arithmatex">\(k\)</span>：</p>
</li>
</ul>
<div class="arithmatex">\[
\text{Sh}_x(k) = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}
\]</div>
<p><img alt="剪切使顶部侧向滑动而底部保持不动" src="../../images/shearing.svg" /></p>
<ul>
<li>
<p>每个点水平滑动 <span class="arithmatex">\(k\)</span> 倍于其高度的距离。当 <span class="arithmatex">\(k = 0.5\)</span> 时，高度为2的点向右移动1。最下面一行保持不动，最上面一行滑动最多。这就是斜体文字的工作原理：正立的字母被剪切，从而向右倾斜。</p>
</li>
<li>
<p>以上所有变换（旋转、缩放、反射、剪切）都是<strong>线性</strong>变换。它们保持原点固定，并保持直线为直线。但<strong>平移</strong>（将所有点按固定量移动）呢？</p>
</li>
<li>
<p>平移<em>不是</em>线性变换，因为它移动了原点。如果将每个点向右移动3，零向量会移动到 <span class="arithmatex">\([3, 0]^T\)</span>，从而破坏了线性性质。为了处理平移，我们使用<strong>仿射变换</strong>，它将线性变换与平移结合起来：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{y} = A\mathbf{x} + \mathbf{t}\]</div>
<ul>
<li>为了将其表示为单个矩阵乘法，我们使用<strong>齐次坐标</strong>：为每个向量添加一个额外的1，并使用一个 <span class="arithmatex">\((n+1) \times (n+1)\)</span> 的矩阵：</li>
</ul>
<div class="arithmatex">\[
\begin{bmatrix} A & \mathbf{t} \\ \mathbf{0}^T & 1 \end{bmatrix} \begin{bmatrix} \mathbf{x} \\ 1 \end{bmatrix} = \begin{bmatrix} A\mathbf{x} + \mathbf{t} \\ 1 \end{bmatrix}
\]</div>
<ul>
<li>
<p>仿射变换保持直线和平行性，但不一定保持角度或长度。电子游戏中的每个物体都使用仿射变换来定位：旋转它、缩放它，然后放置到正确的位置——所有这些都编码在一个矩阵中。</p>
</li>
<li>
<p><strong>退化变换</strong>（奇异矩阵）将空间坍缩到更低维度。</p>
</li>
<li>
<p>例如，矩阵</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}
\]</div>
<p>将每个二维向量映射到一条直线上，因为两列指向同一方向。行列式为零，信息丢失，且该变换不可逆。</p>
<ul>
<li>
<p>将彩色图像（每个像素有3个值：红、绿、蓝）转换为灰度图（每个像素1个值）就是退化变换：颜色信息永久丢失。</p>
</li>
<li>
<p>在机器学习中，线性变换是神经网络的核心。数据被表示为矩阵（向量的堆叠，这些向量代表对象的特征——人、飞机、文本、图像……任何东西！）</p>
</li>
<li>
<p>每一层应用一个矩阵乘法（线性变换），详细内容在其他章节中提供，我们需要解释如何组织这些数据并恰当地引出神经网络。</p>
</li>
<li>
<p>然而，当今最常用的技术几乎完全是将数据通过一系列线性变换传递，我们称之为<strong>Transformer</strong>。</p>
</li>
<li>
<p>Gemini、ChatGPT、Claude、Qwen、DeepSeek以及当今世界上性能最好的AI，都是Transformer！</p>
</li>
</ul>
<h2 id="colabjupyter-notebook">编程练习（使用CoLab或Jupyter Notebook）<a class="headerlink" href="#colabjupyter-notebook" title="Permanent link">&para;</a></h2>
<ol>
<li>对向量应用旋转矩阵，并绘制原始向量和旋转后的向量。尝试不同的角度。</li>
</ol>
<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><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-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="n">theta</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">3</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">R</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="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)],</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="p">[</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</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="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">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">v_rot</span> <span class="o">=</span> <span class="n">R</span> <span class="o">@</span> <span class="n">v</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">plt</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="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">angles</span><span class="o">=</span><span class="s1">&#39;xy&#39;</span><span class="p">,</span> <span class="n">scale_units</span><span class="o">=</span><span class="s1">&#39;xy&#39;</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;original&#39;</span><span class="p">)</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="n">plt</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="n">v_rot</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">v_rot</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">angles</span><span class="o">=</span><span class="s1">&#39;xy&#39;</span><span class="p">,</span> <span class="n">scale_units</span><span class="o">=</span><span class="s1">&#39;xy&#39;</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;blue&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;rotated&#39;</span><span class="p">)</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">)</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span> <span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div>
<ol>
<li>对构成正方形的一组点应用剪切变换，并可视化变形后的形状。</li>
</ol>
<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="n">square</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">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</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">T</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">k</span> <span class="o">=</span> <span class="mf">0.5</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">shear</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="n">k</span><span class="p">],</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">sheared</span> <span class="o">=</span> <span class="n">shear</span> <span class="o">@</span> <span class="n">square</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">square</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">square</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;r-o&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;original&#39;</span><span class="p">)</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sheared</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sheared</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;b-o&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;sheared&#39;</span><span class="p">)</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">();</span> <span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">set_aspect</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div>
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