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<liclass="chapter" data-level="2.7.1" data-path="linear-mappings.html"><ahref="linear-mappings.html#matrix-representation-of-linear-mappings"><iclass="fa fa-check"></i><b>2.7.1</b> Matrix Representation of Linear Mappings</a></li>
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<liclass="chapter" data-level="2.7.2" data-path="linear-mappings.html"><ahref="linear-mappings.html#coordinate-systems-and-bases"><iclass="fa fa-check"></i><b>2.7.2</b> Coordinate Systems and Bases</a></li>
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<liclass="chapter" data-level="2.7.3" data-path="linear-mappings.html"><ahref="linear-mappings.html#basis-change-and-equivalence"><iclass="fa fa-check"></i><b>2.7.3</b> Basis Change and Equivalence</a></li>
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<liclass="chapter" data-level="2.7.4" data-path="linear-mappings.html"><ahref="linear-mappings.html#image-and-kernel-of-a-linear-mapping"><iclass="fa fa-check"></i><b>2.7.4</b> Image and Kernel of a Linear Mapping</a></li>
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<liclass="chapter" data-level="2.7.6" data-path="linear-mappings.html"><ahref="linear-mappings.html#image-and-kernel-of-a-linear-mapping"><iclass="fa fa-check"></i><b>2.7.6</b> Image and Kernel of a Linear Mapping</a></li>
<p>In the machine learning literature, the terms <em>linear</em> and <em>affine</em> are often used interchangeably.</p>
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</div>
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<divclass="definition">
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<p><spanid="def:unlabeled-div-200" class="definition"><strong>Definition 2.36</strong></span>Let <spanclass="math inline">\(V\)</span> be a vector space, <spanclass="math inline">\(x_0 \in V\)</span>, and <spanclass="math inline">\(U \subseteq V\)</span> a subspace. Then
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<p><spanid="def:unlabeled-div-184" class="definition"><strong>Definition 2.35</strong></span>Let <spanclass="math inline">\(V\)</span> be a vector space, <spanclass="math inline">\(x_0 \in V\)</span>, and <spanclass="math inline">\(U \subseteq V\)</span> a subspace. Then
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<spanclass="math display">\[
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L = x_0 + U := \{ x_0 + u : u \in U \}
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\]</span>
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is called an <strong>affine subspace</strong> or <strong>linear manifold</strong> of <spanclass="math inline">\(V\)</span>.</p>
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<ul>
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<li><spanclass="math inline">\(U\)</span> is the <strong>direction space</strong>.<br/>
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</li>
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<li><spanclass="math inline">\(\mathbf{x}_0\)</span> is the <strong>support point</strong>.<br/>
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<li><spanclass="math inline">\(x_0\)</span> is the <strong>support point</strong>.<br/>
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</li>
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<li>If <spanclass="math inline">\(\mathbf{x}_0 \notin U\)</span>, then <spanclass="math inline">\(L\)</span> is <em>not</em> a linear subspace (it does not contain the origin).</li>
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<li>If <spanclass="math inline">\(x_0 \notin U\)</span>, then <spanclass="math inline">\(L\)</span> is <em>not</em> a linear subspace (it does not contain the origin).</li>
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</ul>
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</div>
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<divclass="example">
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<p><spanid="exm:unlabeled-div-201" class="example"><strong>Example 2.59 </strong></span>Points, lines, and planes in <spanclass="math inline">\(\mathbb{R}^3\)</span> that do not necessarily pass through the origin are examples of affine spaces.</p>
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<p><spanid="exm:unlabeled-div-185" class="example"><strong>Example 2.59 </strong></span>Points, lines, and planes in <spanclass="math inline">\(\mathbb{R}^3\)</span> that do not necessarily pass through the origin are examples of affine spaces.</p>
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</div>
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<divclass="definition">
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<p><spanid="def:unlabeled-div-202" class="definition"><strong>Definition 2.37</strong></span>If <spanclass="math inline">\(L = \mathbf{x}_0 + U\)</span> and <spanclass="math inline">\((\mathbf{b}_1, \ldots, \mathbf{b}_k)\)</span> is a basis of <spanclass="math inline">\(U\)</span>,<br/>
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then every <spanclass="math inline">\(\mathbf{x} \in L\)</span> can be written as:
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<p><spanid="def:unlabeled-div-186" class="definition"><strong>Definition 2.36</strong></span>If <spanclass="math inline">\(L = x_0 + U\)</span> and <spanclass="math inline">\((b_1, \ldots, b_k)\)</span> is a basis of <spanclass="math inline">\(U\)</span>,<br/>
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then every <spanclass="math inline">\(x \in L\)</span> can be written as:
<li><p>A line <spanclass="math inline">\(y = x_0 + \lambda b_1\)</span>, where <spanclass="math inline">\(\lambda \in \mathbb{R}\)</span> is an example of a parametric equation in a 1-dimensional affine subspace.</p></li>
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<li><p>A plane <spanclass="math inline">\(y = \mathbf{x}_0 + \lambda_1 \mathbf{b}_1 + \lambda_2 \mathbf{b}_2\)</span>, where <spanclass="math inline">\(\mathbf{b}_1, \mathbf{b}_2\)</span> are linearly independent is an example of a parametric equation in a 2-dimensional affine subspace.</p></li>
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<li><p>A plane <spanclass="math inline">\(y = x_0 + \lambda_1 b_1 + \lambda_2 b_2\)</span>, where <spanclass="math inline">\(b_1, b_2\)</span> are linearly independent is an example of a parametric equation in a 2-dimensional affine subspace.</p></li>
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<li><p>A hyperplane is an example of an <spanclass="math inline">\((n - 1)\)</span>-dimensional affine subspace
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<spanclass="math display">\[
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y = \mathbf{x}_0 + \sum_{i=1}^{n-1} \lambda_i \mathbf{b}_i
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y = x_0 + \sum_{i=1}^{n-1} \lambda_i b_i
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\]</span></p>
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<ul>
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<li>In <spanclass="math inline">\(\mathbb{R}^2\)</span>, a line is a hyperplane.<br/>
<h3><spanclass="header-section-number">2.8.1</span> Relation to Linear Equations<ahref="affine-spaces.html#relation-to-linear-equations" class="anchor-section" aria-label="Anchor link to header"></a></h3>
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<p>For <spanclass="math inline">\(\mathbf{A} \in \mathbb{R}^{m \times n}\)</span> and <spanclass="math inline">\(\mathbf{x} \in \mathbb{R}^m\)</span>, the solution set of <spanclass="math inline">\(\mathbf{A}\lambda = \mathbf{x}\)</span> is either empty or an <strong>affine subspace</strong> of <spanclass="math inline">\(\mathbb{R}^n\)</span> with dimension <spanclass="math inline">\(n - \text{rk}(\mathbf{A})\)</span>.</p>
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<p>For <spanclass="math inline">\(\mathbf{A} \in \mathbb{R}^{m \times n}\)</span> and <spanclass="math inline">\(x \in \mathbb{R}^m\)</span>, the solution set of <spanclass="math inline">\(\mathbf{A}\lambda = x\)</span> is either empty or an <strong>affine subspace</strong> of <spanclass="math inline">\(\mathbb{R}^n\)</span> with dimension <spanclass="math inline">\(n - \text{rk}(\mathbf{A})\)</span>.</p>
<li>The equation <spanclass="math inline">\(\mathbf{A} x = b\)</span> (inhomogeneous system) defines an affine subspace.<br/>
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</li>
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<li>The equation <spanclass="math inline">\(\mathbf{A} \mathbf{x} = \mathbf{0}\)</span> (homogeneous system) defines a vector subspace, which can be seen as a special affine subspace with support point <spanclass="math inline">\(\mathbf{x}_0 = \mathbf{0}\)</span>.</li>
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<li>The equation <spanclass="math inline">\(\mathbf{A} x = 0\)</span> (homogeneous system) defines a <strong>vector subspace</strong>, which can be seen as a special affine subspace with support point <spanclass="math inline">\(x_0 = 0\)</span>.</li>
<h3><spanclass="header-section-number">2.8.2</span> Affine Mappings<ahref="affine-spaces.html#affine-mappings" class="anchor-section" aria-label="Anchor link to header"></a></h3>
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<p>Affine mappings generalize linear mappings by including a <strong>translation</strong>.</p>
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<divclass="definition">
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<p><spanid="def:unlabeled-div-204" class="definition"><strong>Definition 2.38</strong></span>Let <spanclass="math inline">\(V, W\)</span> be vector spaces, <spanclass="math inline">\(\Phi : V \to W\)</span> a linear map, and <spanclass="math inline">\(\mathbf{a} \in W\)</span>. Then
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<p><spanid="def:unlabeled-div-188" class="definition"><strong>Definition 2.37</strong></span>Let <spanclass="math inline">\(V, W\)</span> be vector spaces, <spanclass="math inline">\(\Phi : V \to W\)</span> a linear map, and <spanclass="math inline">\(a \in W\)</span>. Then
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<spanclass="math display">\[
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\varphi : V \to W, \quad x \mapsto \mathbf{a} + \Phi(\mathbf{x})
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\varphi : V \to W, \quad x \mapsto a + \Phi(x)
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\]</span>
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is an <strong>affine mapping</strong> with <strong>translation vector</strong><spanclass="math inline">\(\mathbf{a}\)</span>.</p>
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is an <strong>affine mapping</strong> with <strong>translation vector</strong><spanclass="math inline">\(a\)</span>.</p>
<liclass="chapter" data-level="6.4" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html"><iclass="fa fa-check"></i><b>6.4</b> Summary Statistics and Independence</a>
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<ul>
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<liclass="chapter" data-level="6.4.1" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html#means-and-covariances"><iclass="fa fa-check"></i><b>6.4.1</b> Means and Covariances</a></li>
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<liclass="chapter" data-level="6.4.2" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html#empirical-means-and-covariances"><iclass="fa fa-check"></i><b>6.4.2</b><strong>Empirical Means and Covariances</strong></a></li>
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<liclass="chapter" data-level="6.4.3" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html#alternatice-expressions-for-the-variance"><iclass="fa fa-check"></i><b>6.4.3</b> Alternatice Expressions for the Variance</a></li>
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<liclass="chapter" data-level="6.4.4" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html#sums-and-transformations-of-random-variables"><iclass="fa fa-check"></i><b>6.4.4</b><strong>Sums and Transformations of Random Variables</strong></a></li>
<liclass="chapter" data-level="6.4.6" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html#inner-products-and-geometry-of-random-variables"><iclass="fa fa-check"></i><b>6.4.6</b><strong>Inner Products and Geometry of Random Variables</strong></a></li>
<liclass="chapter" data-level="6.4.2" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html#sums-and-transformations-of-random-variables"><iclass="fa fa-check"></i><b>6.4.2</b><strong>Sums and Transformations of Random Variables</strong></a></li>
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<liclass="chapter" data-level="6.4.3" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html#independence-and-conditional-independence"><iclass="fa fa-check"></i><b>6.4.3</b><strong>Independence and Conditional Independence</strong></a></li>
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<liclass="chapter" data-level="6.4.4" data-path="summary-statistics-and-independence.html"><ahref="summary-statistics-and-independence.html#inner-products-and-geometry-of-random-variables"><iclass="fa fa-check"></i><b>6.4.4</b><strong>Inner Products and Geometry of Random Variables</strong></a></li>
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