阅读翻译Mathematics for Machine Learning之2.8 Affine Subspaces

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First published: 2024-07-24
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2.8 Imitation spaces

Next, we will examine in more detail spaces that are offset from the origin, i.e., spaces that are no longer vector subspaces. In addition, we will briefly discuss the nature of the mappings between these affine spaces, which are analogous to linear mappings.

note. In the machine learning literature, the distinction between linear and affine is sometimes not clear, to the extent that we can find references that refer to affine spaces/mappings as linear spaces/mappings.

2.8.1 Imitation spaces

Definition 2.25(affine subspace). Setting\(V\) is a vector space.\(\boldsymbol{x}_0 \in V\)，\(U \subseteq V\) for a subspace. Then the subset

\[\begin{align*} L & =\boldsymbol{x}_0+U:=\left\{\boldsymbol{x}_0+\boldsymbol{u}: \boldsymbol{u} \in U\right\} \tag{2.130a} \\ & =\left\{\boldsymbol{v} \in V \mid \exists \boldsymbol{u} \in U: \boldsymbol{v}=\boldsymbol{x}_0+\boldsymbol{u}\right\} \subseteq V \tag{2.130b} \end{align*} \]

call sth (by a name)\(V\) (used form a nominal expression)affine subspace (math.)maybeLinear manifolds。\(U\) call sth (by a name)orientationsmaybeDirection space，\(\boldsymbol{x}_0\) call sth (by a name)Support point. In Chapter 12, we refer to such subspaces as hyperplanes.

Note that if the\(\boldsymbol{x}_0 \notin U\), then the definition of affine subspaces excludes\(\mathbf{0}\). Therefore, for\(\boldsymbol{x}_0 \notin U\), affine subspaces are not\(V\) of (linear) subspaces (vector subspaces).

Examples of affine subspaces are\(\mathbb{R}^3\) in points, lines and planes that do not (necessarily) pass through the origin.

note. Consider the vector space\(V\) The two affine subspaces of\(L = \boldsymbol{x}_0 + U\) cap (a poem)\(\tilde{L} = \tilde{\boldsymbol{x}}_0 + \tilde{U}\). If and only if\(U \subseteq \tilde{U}\) both (... and...)\(x_0 - \tilde{x}_0 \in \tilde{U}\) when\(L \subseteq \tilde{L}\)。

Affine subspaces are usually described by parameters: consider a\(V\) (used form a nominal expression)\(k\) dimensional affine space (math.)\(L = \boldsymbol{x}_0 + U\). If\(\left(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_k\right)\) be\(U\) of an ordered base, then each element\(\boldsymbol{x} \in L\) can all be uniquely described as

\[\boldsymbol{x}=\boldsymbol{x}_0+\lambda_1 \boldsymbol{b}_1+\ldots+\lambda_k \boldsymbol{b}_k, \tag{2.131} \]

included among these\(\lambda_1, \ldots, \lambda_k \in \mathbb{R}\). This representation is called having a direction vector\(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_k\) and parameters\(\lambda_1, \ldots, \lambda_k\) (used form a nominal expression)\(L\) The parametric equations of the

**Example 2.26 (affine subspace)**

One-dimensional affine subspaces are called straight lines and can be written\(\boldsymbol{y}=\boldsymbol{x}_0+\lambda \boldsymbol{b}_1\)which\(\lambda \in \mathbb{R}\)，\(U=\operatorname{span}\left[\boldsymbol{b}_1\right] \subseteq \mathbb{R}^n\) be\(\mathbb{R}^n\) of a one-dimensional subspace. This means that the line consists of a pivot point\(\boldsymbol{x}_0\) and a vector defining the direction\(\boldsymbol{b}_1\) Definition. See Figure 2.13 for a schematic.
\(\mathbb{R}^n\) The two-dimensional affine subspace is called the plane. The parametric equation of the plane is\(\boldsymbol{y}=\boldsymbol{x}_0+\lambda_1 \boldsymbol{b}_1+\lambda_2 \boldsymbol{b}_2\)which\(\lambda_1, \lambda_2 \in \mathbb{R}\)，\(U=\operatorname{span}\left[\boldsymbol{b}_1, \boldsymbol{b}_2\right] \subseteq \mathbb{R}^n\). This means that the plane consists of a pivot point\(\boldsymbol{x}_0\) and two linearly independent vectors\(\boldsymbol{b}_1, \boldsymbol{b}_2\) By definition, these two vectors are spread into a direction space (span the direction space).
exist\(\mathbb{R}^n\) Middle.\((n-1)\) The dimensional affine subspace is called a hyperplane and the corresponding parametric equations are\(\boldsymbol{y}=\boldsymbol{x}_0+\sum_{i=1}^{n-1} \lambda_i \boldsymbol{b}_i\)which\(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_{n-1}\) compose\(\mathbb{R}^n\) an\((n-1)\) (math.) dimension subspace (math.)\(U\) of the base. This means that the hyperplane consists of a pivot point\(\boldsymbol{x}_0\) cap (a poem)\((n-1)\) linearly independent vectors\(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_{n-1}\) defined, these vectors are tensored into a direction space. In the\(\mathbb{R}^2\) in which straight lines are also hyperplanes. In the\(\mathbb{R}^3\) in which the plane is also hyperplane.

note(systems of nonchiral linear equations and affine subspaces). For\(\boldsymbol{A} \in \mathbb{R}^{m \times n}\) cap (a poem)\(\boldsymbol{x} \in \mathbb{R}^m\), a system of linear equations\(\boldsymbol{A} \boldsymbol{\lambda}=\boldsymbol{x}\) The solution of is either the empty set or\(\mathbb{R}^n\) The middle dimension is\(n-\operatorname{rk}(\boldsymbol{A})\) of affine projective subspaces. In particular, when\(\left(\lambda_1, \ldots, \lambda_n\right) \neq (0, \ldots, 0)\) When the linear equation\(\lambda_1 \boldsymbol{b}_1 + \ldots + \lambda_n \boldsymbol{b}_n = \boldsymbol{x}\) The solution of\(\mathbb{R}^n\) A hyperplane in the

exist\(\mathbb{R}^n\) In each of the\(k\) Dimensional affine subspaces are all systems of nonchiral linear equations\(\boldsymbol{A x}=\boldsymbol{b}\) The solution to the problem, where\(\boldsymbol{A} \in \mathbb{R}^{m \times n}\)，\(\boldsymbol{b} \in \mathbb{R}^m\) besides\(\operatorname{rk}(\boldsymbol{A})=n-k\). Recall that for the system of chi-square equations\(\boldsymbol{A x}=\mathbf{0}\), the solution is a vector subspace, which we can also consider as a special affine space with the branch point\(\boldsymbol{x}_0=\mathbf{0}\)。

2.8.2 Affine Mapping

Similar to the linear mapping between vector spaces we discussed in Section 2.7, we can define an affine mapping between two affine spaces. Linear mappings and affine mappings are closely related. Thus, many of the properties that we already know from linear mappings, such as the fact that the composite of a linear mapping is a linear mapping, also apply to affine mappings.

Definition 2.26(affine mapping). For two vector spaces\(V, W\)A linear mapping\(\Phi: V \rightarrow W\)as well as\(\boldsymbol{a} \in W\)Mapping

\[\begin{align*} \phi: V & \rightarrow W \tag{2.132} \\ \boldsymbol{x} & \mapsto \boldsymbol{a} + \Phi(\boldsymbol{x}) \tag{2.133} \end{align*} \]

is from\(V\) until (a time)\(W\) The affine mapping of the Vector\(\boldsymbol{a}\) be known as\(\phi\) The translation vector of the

Every affine mapping\(\phi: V \rightarrow W\) is also a linear mapping\(\Phi: V \rightarrow W\) cap (a poem)\(W\) The translation in the\(\tau: W \rightarrow W\) The composite that makes the\(\phi = \tau \circ \Phi\)The Mapping\(\Phi\) cap (a poem)\(\tau\) is uniquely determined.
affine map (math.)\(\phi: V \rightarrow W, \phi^{\prime}: W \rightarrow X\) complex\(\phi^{\prime} \circ \phi\) It's faux-shot.
in the event that\(\phi\) are bijective, affine mappings keep the geometry invariant. They also preserve dimensionality and parallelism.