阅读翻译Mathematics for Machine Learning之2.7 Linear Mappings

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2.7 Linear Mapping

In what follows, we will look at mappings that maintain the structure of vector spaces, which will allow us to define the notion of coordinates. At the beginning of this chapter, we mentioned that vectors are objects that can be added and multiplied by scalars and the result is still a vector. When applying mappings, we want to maintain this property: consider two real vector spaces\(V, W\). If the mapping\(\Phi: V \rightarrow W\) satisfies the following conditions, then it maintains the structure of the vector space:

\[\begin{align*} \Phi(\boldsymbol{x}+\boldsymbol{y}) & =\Phi(\boldsymbol{x})+\Phi(\boldsymbol{y}) \tag{2.85} \\ \Phi(\lambda \boldsymbol{x}) & =\lambda \Phi(\boldsymbol{x}) \tag{2.86} \end{align*} \]

For all\(\boldsymbol{x}, \boldsymbol{y} \in V\) cap (a poem)\(\lambda \in \mathbb{R}\) Established. We can summarize this with the following definition:

Definition 2.15(linear mapping). For vector spaces\(V, W\)A mapping\(\Phi: V \rightarrow W\) be known aslinear map (math.)(orvector space homomorphism/linear transformation), if

\[\forall \boldsymbol{x}, \boldsymbol{y} \in V \ \forall \lambda, \psi \in \mathbb{R}: \Phi(\lambda \boldsymbol{x}+\psi \boldsymbol{y})=\lambda \Phi(\boldsymbol{x})+\psi \Phi(\boldsymbol{y}) \tag{2.87} \]

The results show that we can represent linear mappings as matrices (see Section 2.7.1). Recall that we can also use a set of vectors as columns of a matrix. When working with matrices, we must remember what the matrix represents: a linear mapping or a collection of vectors. We will discuss linear mappings in more detail in Chapter 4. Before we continue, we will briefly describe some special mappings.

Definition 2.16(single shot, full shot, double shot). Consider a mapping\(\Phi\) : \(\mathcal{V} \rightarrow \mathcal{W}\)which\(\mathcal{V}\) cap (a poem)\(\mathcal{W}\) can be any set. So\(\Phi\) Known as:

Injective, if\(\forall \boldsymbol{x}, \boldsymbol{y} \in \mathcal{V}\)Yes\(\Phi(\boldsymbol{x})=\Phi(\boldsymbol{y}) \Longrightarrow \boldsymbol{x}=\boldsymbol{y}\)。
Surjective if\(\Phi(\mathcal{V})=\mathcal{W}\)。
Bijective (Bijective) if it is both a single shot and a full shot.

in the event that\(\Phi\) It's a full shot, then.\(\mathcal{W}\) Each element in the\(\Phi\) through (a gap)\(\mathcal{V}\) In "arrive". bijection\(\Phi\) can be "inverted", i.e., there exists a mapping\(\Psi\) : \(\mathcal{W} \rightarrow \mathcal{V}\) feasible\(\Psi \circ \Phi(\boldsymbol{x})=\boldsymbol{x}\). This mapping\(\Psi\) be known as\(\Phi\) The inverse mapping, usually denoted as\(\Phi^{-1}\)。

With these definitions, we introduce the following vector spaces\(V\) cap (a poem)\(W\) A special case of a linear mapping between the

Isomorphism:\(\Phi: V \rightarrow W\) Linear and bijective
Endomorphism:\(\Phi: V \rightarrow V\) biometrics
Automorphism:\(\Phi: V \rightarrow V\) Linear and bijective
We define\(\operatorname{id}_V: V \rightarrow V, \boldsymbol{x} \mapsto \boldsymbol{x}\) because of\(V\) in the constant mapping or constant self-isomorphism.

**Example 2.19 (Homomorphism)**

map (math.)\(\Phi: \mathbb{R}^2 \rightarrow \mathbb{C}, \Phi(\boldsymbol{x})=x_1+i x_2\) is a homomorphism:

\[\begin{aligned} \Phi\left(\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right]+\left[\begin{array}{l} y_1 \\ y_2 \end{array}\right]\right) & =\left(x_1+y_1\right)+i\left(x_2+y_2\right)=x_1+i x_2+y_1+i y_2 \\ & =\Phi\left(\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right]\right)+\Phi\left(\left[\begin{array}{l} y_1 \\ y_2 \end{array}\right]\right) \\ \Phi\left(\lambda\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right]\right) & =\lambda x_1+\lambda i x_2=\lambda\left(x_1+i x_2\right)=\lambda \Phi\left(\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right]\right) . \end{aligned} \tag{2.88} \]

This also explains why the plural can be represented as\(\mathbb{R}^2\) in the tuple: there exists a bi-rayability map that can be\(\mathbb{R}^2\) The element-by-element addition of a tuple in the center is converted to the set of complex numbers corresponding to the addition. Note that we show only linearity here, not bijection.

Theorem 2.17(Theorem 3.59 of Axler (2015)). Finite-dimensional vector spaces\(V\) cap (a poem)\(W\) beisomorphic, if and only if\(\operatorname{dim}(V)=\operatorname{dim}(W)\)。

Theorem 2.17 shows that there exists a linear, bijective map between two vector spaces of the same dimension. Intuitively, this implies that vector spaces of the same dimension are somehow identical, since they can be transformed into each other without suffering any loss.

Theorem 2.17 also provides us with the ability to convert\(\mathbb{R}^{m \times n}\)（\(m \times n\) vector space of matrices) and\(\mathbb{R}^{mn}\)(length\(mn\) vector space of vectors) are regarded as identical for the same reason, since their dimensions are both\(mn\), and there exists a linear, bijective mapping transforming one into the other.

note. Consider the vector space\(V, W, X\). Then:

For linear mappings\(\Phi: V \rightarrow W\) cap (a poem)\(\Psi: W \rightarrow X\)Mapping\(\Psi \circ \Phi: V \rightarrow X\) It's also linear.
in the event that\(\Phi: V \rightarrow W\) is isomorphism, then\(\Phi^{-1}: W \rightarrow V\) It is also isomorphic.
in the event that\(\Phi: V \rightarrow W, \Psi: V \rightarrow W\) is linear, then\(\Phi+\Psi\) cap (a poem)\(\lambda \Phi, \lambda \in \mathbb{R}\), which is also linear.

2.7.1 Matrix Representation of Linear Mappings

whichever\(n\) dimensional vector spaces are all associated with\(\mathbb{R}^n\) isomorphism (Theorem 2.17). We consider a\(n\) dimensional vector space\(V\) exclusionary rule\(\left\{\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right\}\). In what follows, the order of the basis vectors is important. Therefore, we write

\[B=\left(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right) \tag{2.89} \]

and that this\(n\) The tuple is\(V\) of the ordered base.

note(Symbols). The symbols we're using are a bit complicated, so we'll summarize some parts here.\(B=\left(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right)\) is an ordered base.\(\mathcal{B}=\left\{\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right\}\) is a (disordered) basis.\(\boldsymbol{B}=\left[\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right]\) is a matrix whose columns are vectors\(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\)。

Definition 2.18(coordinates). Consider a vector space\(V\) and its ordered group\(B=\left(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right)\). For any\(\boldsymbol{x} \in V\), we can obtain a unique representation (linear combination)

\[\boldsymbol{x}=\alpha_1 \boldsymbol{b}_1+\ldots+\alpha_n \boldsymbol{b}_n \]

included among these\(\alpha_1, \ldots, \alpha_n\) be\(\boldsymbol{x}\) as opposed to\(B\) coordinates, and the vector

\[\boldsymbol{\alpha}=\left[\begin{array}{c} \alpha_1 \\ \vdots \\ \alpha_n \end{array}\right] \in \mathbb{R}^n \]

be\(\boldsymbol{x}\) As opposed to the ordered basis\(B\) (used form a nominal expression)coordinate vector/coordinate representation。

A basis actually defines a coordinate system. The familiar two-dimensional Cartesian coordinate system consists of the standard basis vectors\(\boldsymbol{e}_1, \boldsymbol{e}_2\) Zhang Cheng's. In this coordinate system, the vector\(\boldsymbol{x} \in \mathbb{R}^2\) There is a representation that tells us how to linearly combine\(\boldsymbol{e}_1\) cap (a poem)\(\boldsymbol{e}_2\) to get\(\boldsymbol{x}\). However.\(\mathbb{R}^2\) of any basis defines a valid coordinate system, and the same vector\(\boldsymbol{x}\) primary legislation\(\left(\boldsymbol{b}_1, \boldsymbol{b}_2\right)\) There may be different coordinate representations. In Figure 2.8, the vector\(\boldsymbol{x}\) Relative to standard base\(\left(\boldsymbol{e}_1, \boldsymbol{e}_2\right)\) The coordinates of\([2,2]^{\top}\). However, relative to the base\(\left(\boldsymbol{b}_1, \boldsymbol{b}_2\right)\)vectors\(\boldsymbol{x}\) denoted by\([1.09,0.72]^{\top}\)namely\(\boldsymbol{x}=1.09 \boldsymbol{b}_1+0.72 \boldsymbol{b}_2\). In the next section, we will explore how to obtain this representation.

**Example 2.20**

Let's look at a geometric vector\(\boldsymbol{x} \in \mathbb{R}^2\)Its relative\(\mathbb{R}^2\) standardized base\(\left(\boldsymbol{e}_1, \boldsymbol{e}_2\right)\) The coordinates of the\([2,3]^{\top}\). This means that we can write\(\boldsymbol{x}=2 \boldsymbol{e}_1+3 \boldsymbol{e}_2\). However, we do not have to choose a standard basis to represent this vector. If we use the basis vector\(\boldsymbol{b}_1=[1,-1]^{\top}\) cap (a poem)\(\boldsymbol{b}_2=[1,1]^{\top}\)We will get the coordinates\(\frac{1}{2}[-1,5]^{\top}\) to denote the difference between the\(\left(\boldsymbol{b}_1, \boldsymbol{b}_2\right)\) of the same vector (see Figure 2.9).

note. For a\(n\) dimensional vector space\(V\) cap (a poem)\(V\) An ordered group of\(B\)Mapping\(\Phi: \mathbb{R}^n \rightarrow V, \Phi\left(\boldsymbol{e}_i\right)=\boldsymbol{b}_i, i=1, \ldots, n\) is linear (and by Theorem 2.17, an isomorphism), where\(\left(\boldsymbol{e}_1, \ldots, \boldsymbol{e}_n\right)\) be\(\mathbb{R}^n\) The standard base of the

We are now ready to establish explicitly the connection between linear mappings between matrices and finite-dimensional vector spaces.

Definition 2.19(transformation matrix). Consider the vector space\(V\) cap (a poem)\(W\), they each have corresponding (ordered) bases\(B=\left(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right)\) cap (a poem)\(C=\left(\boldsymbol{c}_1, \ldots, \boldsymbol{c}_m\right)\). Furthermore, we consider a linear mapping\(\Phi: V \rightarrow W\). For\(j \in\{1, \ldots, n\}\)，

\[\Phi\left(\boldsymbol{b}_j\right)=\alpha_{1 j} \boldsymbol{c}_1+\cdots+\alpha_{m j} \boldsymbol{c}_m=\sum_{i=1}^m \alpha_{i j} \boldsymbol{c}_i \tag{2.92} \]

be\(\Phi\left(\boldsymbol{b}_j\right)\) as opposed to\(C\) of the unique representation. We then call\(m \times n\) matrices\(\boldsymbol{A}_{\Phi}\), whose elements are given by the following equation

\[A_{\Phi}(i, j)=\alpha_{i j}, \tag{2.93} \]

because of\(\Phi\) (used form a nominal expression)transformation matrix(as opposed to\(V\) ordered group (math.)\(B\) cap (a poem)\(W\) ordered group (math.)\(C\)）。

\(\Phi\left(\boldsymbol{b}_j\right)\) as opposed to\(W\) ordered group (math.)\(C\) The coordinates of\(\boldsymbol{A}_{\Phi}\) thirteenth meeting of the Conference of the Parties to the Convention on Biological Diversity (CBD)\(j\) Column. Consider the (finite-dimensional) vector space\(V, W\) having an ordered basis\(B, C\) and linear mappings\(\Phi: V \rightarrow W\) and its transformation matrix\(\boldsymbol{A}_{\Phi}\). If\(\hat{\boldsymbol{x}}\) be\(\boldsymbol{x} \in V\) as opposed to\(B\) The coordinate vectors of\(\hat{\boldsymbol{y}}\) be\(\boldsymbol{y}=\Phi(\boldsymbol{x}) \in W\) as opposed to\(C\) vector of coordinates, then

\[\hat{\boldsymbol{y}}=\boldsymbol{A}_{\Phi} \hat{\boldsymbol{x}} . \tag{2.94} \]

This means that the transformation matrix can be used to set the transformation matrix with respect to\(V\) The coordinates of the ordered bases in are mapped to coordinates relative to the base of the\(W\) coordinates of the ordered bases in the

**Example 2.21 (transformation matrix)**

Consider a homomorphism\(\Phi: V \rightarrow W\) as well as\(V\) ordered group (math.)\(B=\left(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_3\right)\) cap (a poem)\(W\) ordered group (math.)\(C=\left(\boldsymbol{c}_1, \ldots, \boldsymbol{c}_4\right)\)The Given

\[\begin{aligned} & \Phi\left(b_1\right)=c_1-c_2+3 c_3-c_4, \\ & \Phi\left(b_2\right)=2 c_1+c_2+7 c_3+2 c_4, \\ & \Phi\left(b_3\right)=3 c_2+c_3+4 c_4, \end{aligned} \tag{2.95} \]

as opposed to\(B\) cap (a poem)\(C\) transformation matrix of a function (math.)\(\boldsymbol{A}_{\Phi}\) fulfillment\(\Phi\left(\boldsymbol{b}_k\right)=\sum_{i=1}^4 \alpha_{i k} \boldsymbol{c}_i, k=1, \ldots, 3\)which is expressed as

\[\boldsymbol{A}_{\Phi}=\left[\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2, \boldsymbol{\alpha}_3\right]=\left[\begin{array}{ccc} 1 & 2 & 0 \\ -1 & 1 & 3 \\ 3 & 7 & 1 \\ -1 & 2 & 4 \end{array}\right], \tag{2.96} \]

included among these\(\boldsymbol{\alpha}_j, j=1,2,3\)Yes.\(\Phi\left(\boldsymbol{b}_j\right)\) as opposed to\(C\) The coordinate vector of the

Figure 2.10 gives three examples of linear transformations of a set of vectors. Figure 2.10(a) illustrates\(\mathbb{R}^2\) The 400 vectors in the\((x_1, x_2)\) The coordinates are represented by a point at the coordinates. These vectors are arranged in a square. When we use the matrix\(\boldsymbol{A}_1\)(in Eq. 2.97) When applying a linear transformation to each of these vectors, we obtain the rotated square in Fig. 2.10(b). If we apply the linear transformation given by\(\boldsymbol{A}_2\) representation of the linear mapping, we will obtain the rectangle in Fig. 2.10(c) where each\(x_1\) The coordinates are stretched by a factor of 2. Figure 2.10(d) illustrates the use of the\(\boldsymbol{A}_3\) The original square after performing a linear transformation, which combines the effects of reflection, rotation and stretching.

2.7.2 Fundamental transformations

In what follows, we will look more closely at linear mappings\(\Phi: V \rightarrow W\) The transformation matrix of the\(V\) cap (a poem)\(W\) How does the base time in the Consider\(V\) The two ordered groups of

\[B=\left(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right), \quad \tilde{B}=\left(\tilde{\boldsymbol{b}}_1, \ldots, \tilde{\boldsymbol{b}}_n\right) \tag{2.98} \]

cap (a poem)\(W\) The two ordered groups of

\[C=\left(\boldsymbol{c}_1, \ldots, \boldsymbol{c}_m\right), \quad \tilde{C}=\left(\tilde{\boldsymbol{c}}_1, \ldots, \tilde{\boldsymbol{c}}_m\right) \tag{2.99} \]

In addition.\(A_{\Phi} \in \mathbb{R}^{m \times n}\) is relative to the base\(B\) cap (a poem)\(C\) linear mapping of\(\Phi: V \rightarrow W\) The transformation matrix of the\(\tilde{\boldsymbol{A}}_{\Phi} \in \mathbb{R}^{m \times n}\) is the same as\(\tilde{B}\) cap (a poem)\(\tilde{C}\) of the corresponding transformation matrices. Next, we will study the\(\boldsymbol{A}\) cap (a poem)\(\tilde{\boldsymbol{A}}\) is how it is related, i.e., if we choose to start from the\(B, C\) modifier\(\tilde{B}, \tilde{C}\), can we/how can we incorporate\(\boldsymbol{A}_{\Phi}\) convert to\(\tilde{A}_{\Phi}\)。

note. We actually get the constant mapping\(\mathrm{id}_V\) of different coordinate representations. In the context of Figure 2.9, this means that without changing the vector\(\boldsymbol{x}\) The case will be relative to the\(\left(\boldsymbol{e}_1, \boldsymbol{e}_2\right)\) is mapped to coordinates relative to the\(\left(b_1, b_2\right)\) of the coordinates. By changing the basis and accordingly the representation of the vectors, the transformation matrix with respect to this new basis can have a particularly simple form, which permits a straightforward computation.

**Example 2.23 (base transformations)**

Consider an object that is relative to the\(\mathbb{R}^2\) The transformation matrix of the standard basis in

\[\boldsymbol{A}=\left[\begin{array}{ll} 2 & 1 \\ 1 & 2 \end{array}\right] \tag{2.100} \]

If we define a new base

\[B=\left(\left[\begin{array}{l} 1 \\ 1 \end{array}\right],\left[\begin{array}{c} 1 \\ -1 \end{array}\right]\right) \tag{2.101} \]

We will obtain a diagonal transformation matrix

\[\tilde{\boldsymbol{A}}=\left[\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right] \tag{2.102} \]

relative to the base\(B\)Its ratio\(\boldsymbol{A}\) Easier to handle.

Next, we will study the mapping that transforms a coordinate vector under one base to a coordinate vector under another base. We will first state the main results and then provide explanations.

Theorem 2.20(base transform). For linear mappings\(\Phi: V \rightarrow W\)，\(V\) ordered group (math.)

\[B=\left(\boldsymbol{b}_1, \ldots, \boldsymbol{b}_n\right), \quad \tilde{B}=\left(\tilde{\boldsymbol{b}}_1, \ldots, \tilde{\boldsymbol{b}}_n\right) \tag{2.103} \]

cap (a poem)\(W\) ordered group (math.)

\[C=\left(\boldsymbol{c}_1, \ldots, \boldsymbol{c}_m\right), \quad \tilde{C}=\left(\tilde{\boldsymbol{c}}_1, \ldots, \tilde{\boldsymbol{c}}_m\right) \tag{2.104} \]

and relative to the base\(B\) cap (a poem)\(C\) (used form a nominal expression)\(\Phi\) transformation matrix of a function (math.)\(\boldsymbol{A}_{\Phi}\)Relative to the base\(\tilde{B}\) cap (a poem)\(\tilde{C}\) The corresponding transformation matrix of\(\tilde{A}_{\Phi}\) is given by the following equation:

\[\tilde{A}_{\Phi}=\boldsymbol{T}^{-1} \boldsymbol{A}_{\Phi} S \tag{2.105} \]

Here.\(S \in \mathbb{R}^{n \times n}\) is the value that will be set relative to the\(\tilde{B}\) is mapped to coordinates relative to the\(B\) of the coordinates of the\(\mathrm{id}_V\) The transformation matrix of the\(\boldsymbol{T} \in \mathbb{R}^{m \times m}\) is the value that will be set relative to the\(\tilde{C}\) is mapped to coordinates relative to the\(C\) of the coordinates of the\(\mathrm{id}_W\) of the transformation matrix.

show that Following the approach of Drumm and Weil (2001), we can place the\(V\) new base\(\tilde{B}\) The vector representation of the base\(B\) A linear combination of the basis vectors of

\[\tilde{\boldsymbol{b}}_j=s_{1 j} \boldsymbol{b}_1+\cdots+s_{n j} \boldsymbol{b}_n=\sum_{i=1}^n s_{i j} \boldsymbol{b}_i, \quad j=1, \ldots, n . \tag{2.106} \]

Similarly, we will\(W\) The new basis vectors of\(\tilde{C}\) denoted by\(C\) A linear combination of the basis vectors of

\[\tilde{\boldsymbol{c}}_k=t_{1 k} \boldsymbol{c}_1+\cdots+t_{m k} \boldsymbol{c}_m=\sum_{l=1}^m t_{l k} \boldsymbol{c}_l, \quad k=1, \ldots, m . \tag{2.107} \]

We define\(\boldsymbol{S} = (s_{ij}) \in \mathbb{R}^{n \times n}\) is the transformation matrix which will be relative to\(\tilde{B}\) is mapped to coordinates relative to the\(B\) coordinates, define the\(\boldsymbol{T} = (t_{lk}) \in \mathbb{R}^{m \times m}\) is the transformation matrix which will be relative to\(\tilde{C}\) is mapped to coordinates relative to the\(C\) of the coordinates. In particular.\(\boldsymbol{S}\) thirteenth meeting of the Conference of the Parties to the Convention on Biological Diversity (CBD)\(j\) columns\(\tilde{\boldsymbol{b}}_j\) as opposed to\(B\) The coordinates of the representation of the\(\boldsymbol{T}\) thirteenth meeting of the Conference of the Parties to the Convention on Biological Diversity (CBD)\(k\) columns\(\tilde{\boldsymbol{c}}_k\) as opposed to\(C\) The coordinates of the representation. Note that the\(\boldsymbol{S}\) cap (a poem)\(\boldsymbol{T}\) are regular matrices.

We will look at it from two perspectives\(\Phi(\tilde{\boldsymbol{b}}_j)\). First, application mapping\(\Phi\)We get that for all\(j=1, \ldots, n\)：

where we first combine the new basis vectors\(\tilde{\boldsymbol{c}}_k \in W\) is denoted as the basis vector\(\boldsymbol{c}_l \in W\) of linear combinations and then swap the order of summation.

Alternatively, when we set the\(\tilde{\boldsymbol{b}}_j \in V\) denoted by\(\boldsymbol{b}_j \in V\) When the linear combination of

Here, we utilize the\(\Phi\) The linear characterization of the Comparison of Eqs. (2.108) and (2.109b) leads to the conclusion that for all the\(j=1, \ldots, n\) cap (a poem)\(l=1, \ldots, m\) there are

\[\sum_{k=1}^m t_{l k} \tilde{a}_{k j}=\sum_{i=1}^n a_{l i} s_{i j} \tag{2.110} \]

Therefore.

\[\boldsymbol{T} \tilde{\boldsymbol{A}}_{\Phi}=\boldsymbol{A}_{\Phi} \boldsymbol{S} \in \mathbb{R}^{m \times n}, \tag{2.111} \]

So there's

\[\tilde{A}_{\Phi}=T^{-1} A_{\Phi} S, \tag{2.112} \]

This proves Theorem 2.20.

Theorem 2.20 tells us that when\(V\) The base in (\(B\) (indicates passive-voice clauses)\(\tilde{B}\) (superseded) and\(W\) The base in (\(C\) (indicates passive-voice clauses)\(\tilde{C}\) substitution) changes when the linear mapping\(\Phi: V \rightarrow W\) transformation matrix of a function (math.)\(\boldsymbol{A}_{\Phi}\) is replaced by the equivalent matrix\(\tilde{\boldsymbol{A}}_{\Phi}\), whose relationship is:

\[\tilde{A}_{\Phi}=T^{-1} A_{\Phi} S . \tag{2.113} \]

Figure 2.11 illustrates this relationship: consider a homomorphic mapping\(\Phi: V \rightarrow W\) up to\(V\) ordered group (math.)\(B, \tilde{B}\) cap (a poem)\(W\) ordered group (math.)\(C, \tilde{C}\)The Mapping\(\Phi_{C B}\) be\(\Phi\) An instance of the\(B\) The basis vectors of the mapping to\(C\) of linear combinations of basis vectors. Suppose we know that\(\Phi_{C B}\) transformation matrix of a function (math.)\(\boldsymbol{A}_{\Phi}\)which corresponds to the ordered basis\(B, C\). When we are in the\(V\) choose from\(B\) until (a time)\(\tilde{B}\) and in\(W\) choose from\(C\) until (a time)\(\tilde{C}\) When performing the basis transformation, we can determine the corresponding transformation matrix\(\tilde{\boldsymbol{A}}_{\Phi}\) as below

First, we find the linear mapping\(\Psi_{B \tilde{B}}: V \rightarrow V\) The matrix representation of the mapping will be relative to the new base\(\tilde{B}\) The coordinates are mapped (uniquely) with respect to the "old" base\(B\) The coordinates (in\(V\) (center).
Then, we use the\(\Phi_{C B}: V \rightarrow W\) transformation matrix of a function (math.)\(\boldsymbol{A}_{\Phi}\) Mapping these coordinates to\(W\) relative to\(C\) The coordinates of the
Finally, we use the linear mapping\(\Xi_{\tilde{C} C}: W \rightarrow W\) would be relative to\(C\) is mapped to coordinates relative to the\(\tilde{C}\) of the coordinates. Therefore, we can put the linear mapping\(\Phi_{\tilde{C} \tilde{B}}\) is expressed as a combination of linear mappings involving "old" bases:

\[\Phi_{\tilde{C} \tilde{B}}=\Xi_{\tilde{C} C} \circ \Phi_{C B} \circ \Psi_{B \tilde{B}}=\Xi_{C \tilde{C}}^{-1} \circ \Phi_{C B} \circ \Psi_{B \tilde{B}} . \tag{2.114} \]

Specifically, we use\(\Psi_{B \tilde{B}}=\operatorname{id}_V\) cap (a poem)\(\Xi_{C \tilde{C}}=\mathrm{id}_W\), i.e., a constant map that maps vectors to themselves, but with respect to different bases.

Figure 2.11 For homomorphic mapping\(\Phi: V \rightarrow W\) as well as\(V\) ordered group (math.)\(B, \tilde{B}\) cap (a poem)\(W\) ordered group (math.)\(C, \tilde{C}\)(marked in blue), we can set the base with respect to the\(\tilde{B}, \tilde{C}\) mapping\(\Phi_{\tilde{C} \tilde{B}}\) is equivalently expressed as a homomorphic mapping\(\Phi_{\tilde{C} \tilde{B}}=\) \(\Xi_{\tilde{C} C} \circ \Phi_{C B} \circ \Psi_{B \tilde{B}}\) combinations whose subscripts indicate the corresponding bases. The corresponding transformation matrices are labeled in red.

Definition 2.21(Equivalent). If there exists a regular matrix\(S \in \mathbb{R}^{n \times n}\) cap (a poem)\(\boldsymbol{T} \in \mathbb{R}^{m \times m}\)makes\(\tilde{A} = T^{-1} A S\), then the two matrices\(\boldsymbol{A}, \tilde{A} \in \mathbb{R}^{m \times n}\) beequivalent。

Definition 2.22(similar). If there exists a regular matrix\(S \in \mathbb{R}^{n \times n}\) feasible\(\tilde{A} = \boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S}\), then the two matrices\(\boldsymbol{A}, \tilde{A} \in \mathbb{R}^{n \times n}\) beakin。

note. Similar matrices are always equivalent. However, equivalent matrices are not necessarily similar.

note. Consider the vector space\(V, W, X\). We already know from the Remark after Theorem 2.17 that for linear mappings\(\Phi: V \rightarrow W\) cap (a poem)\(\Psi: W \rightarrow X\)Mapping\(\Psi \circ \Phi: V \rightarrow X\) is also linear. For the transformation matrix of the corresponding mapping\(\boldsymbol{A}_{\Phi}\) cap (a poem)\(\boldsymbol{A}_{\Psi}\)and the overall transformation matrix is\(\boldsymbol{A}_{\Psi \circ \Phi} = \boldsymbol{A}_{\Psi} \boldsymbol{A}_{\Phi}\)。

According to this note, we can view the basis transformations in terms of combinatorial linear mappings:

\(\boldsymbol{A}_{\Phi}\) is relative to the base\(B, C\) linear mapping of\(\Phi_{C B}: V \rightarrow W\) of the transformation matrix.
\(\tilde{A}_{\Phi}\) is relative to the base\(\tilde{B}, \tilde{C}\) linear mapping of\(\Phi_{\tilde{C} \tilde{B}}: V \rightarrow W\) of the transformation matrix.
\(S\) is relative to the base\(B, \tilde{B}\) linear mapping of\(\Psi_{B \tilde{B}}: V \rightarrow V\)(self-isomorphic) transformation matrices, which are represented by the\(B\) to indicate\(\tilde{B}\). Usually.\(\Psi=\mathrm{id}_V\) be\(V\) in the constant mapping.
\(\boldsymbol{T}\) is relative to the base\(C, \tilde{C}\) linear mapping of\(\Xi_{C \tilde{C}}: W \rightarrow W\)(self-isomorphic) transformation matrices, which are represented by the\(C\) to indicate\(\tilde{C}\). Usually.\(\Xi=\mathrm{id}_W\) be\(W\) in the constant mapping.

If we (informally) write down these transformations only in terms of bases, then\(\boldsymbol{A}_{\Phi}: B \rightarrow C, \tilde{\boldsymbol{A}}_{\Phi}: \tilde{B} \rightarrow \tilde{C}, \boldsymbol{S}: \tilde{B} \rightarrow B, \boldsymbol{T}: \tilde{C} \rightarrow C\) cap (a poem)\(\boldsymbol{T}^{-1}: C \rightarrow \tilde{C}\)furthermore

\[\begin{align*} \tilde{B} \rightarrow \tilde{C} & =\tilde{B} \rightarrow B \rightarrow C \rightarrow \tilde{C} \tag{2.115} \\ \tilde{\boldsymbol{A}}_{\Phi} & =\boldsymbol{T}^{-1} \boldsymbol{A}_{\Phi} \boldsymbol{S} .\tag{2.116} \end{align*} \]

Note that the order of execution in Eq. (2.116) is right-to-left, since the vectors are multiplied on the right-hand side, so the\(\boldsymbol{x} \mapsto \boldsymbol{S} \boldsymbol{x} \mapsto \boldsymbol{A}_{\Phi}(\boldsymbol{S} \boldsymbol{x}) \mapsto T^{-1}\left(A_{\Phi}(S x)\right)=\tilde{\boldsymbol{A}}_{\Phi} x\)。

**Example 2.24 (base transformations)**

Consider a linear mapping\(\Phi: \mathbb{R}^3 \rightarrow \mathbb{R}^4\)and its transformation matrix is

\[\boldsymbol{A}_{\Phi}=\left[\begin{array}{ccc} 1 & 2 & 0 \\ -1 & 1 & 3 \\ 3 & 7 & 1 \\ -1 & 2 & 4 \end{array}\right] \tag{2.117} \]

Relative to standard base

\[B=\left(\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right]\right), \quad C=\left(\left[\begin{array}{l} 1 \\ 0 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 0 \\ 0 \\ 1 \end{array}\right]\right) . \tag{2.118} \]

We need to find out what is happening relative to the new base

\[\tilde{B}=\left(\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]\right) \in \mathbb{R}^3, \quad \tilde{C}=\left(\left[\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{l} 1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 1 \\ 0 \\ 0 \\ 1 \end{array}\right]\right) . \tag{2.119} \]

The transformation matrix under\(\tilde{\boldsymbol{A}}_{\Phi}\)。

imitate

\[\boldsymbol{S}=\left[\begin{array}{lll} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right], \quad \boldsymbol{T}=\left[\begin{array}{llll} 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \tag{2.120} \]

included among these\(S\) thirteenth meeting of the Conference of the Parties to the Convention on Biological Diversity (CBD)\(i\) columns\(\tilde{\boldsymbol{b}}_i\) relative to the base\(B\) The coordinate representation of the Since the\(B\) are standard bases and the coordinate representations are easy to find. For the general basis\(B\)We need to solve the system of linear equations in order to find\(\lambda_i\) feasible\(\sum_{i=1}^3 \lambda_i \boldsymbol{b}_i=\tilde{\boldsymbol{b}}_j, j=1, \ldots, 3\). Similarly.\(T\) thirteenth meeting of the Conference of the Parties to the Convention on Biological Diversity (CBD)\(j\) columns\(\tilde{c}_j\) relative to the base\(C\) The coordinates of the representation.

As a result, we get

\[\begin{align*} \tilde{\boldsymbol{A}}_{\Phi} & =\boldsymbol{T}^{-1} \boldsymbol{A}_{\Phi} \boldsymbol{S}=\frac{1}{2}\left[\begin{array}{cccc} 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ -1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right]\left[\begin{array}{ccc} 3 & 2 & 1 \\ 0 & 4 & 2 \\ 10 & 8 & 4 \\ 1 & 6 & 3 \end{array}\right] \tag{2.121a} \\ & =\left[\begin{array}{ccc} -4 & -4 & -2 \\ 6 & 0 & 0 \\ 4 & 8 & 4 \\ 1 & 6 & 3 \end{array}\right] . \tag{2.121b} \end{align*} \]

In Chapter 4, we will be able to use the notion of basis transformations to find a basis that gives a particularly simple (diagonal) form to the transformation matrix of a self-homomorphism. In Chapter 10, we will study a data compression problem and find a convenient basis onto which we can project data while minimizing the compression loss.

3.7.3 Images and nuclei

The image and kernel of a linear map are vector subspaces with certain important properties. Next, we will describe them more carefully.

Definition 2.23(like and nuclear).

insofar as\(\Phi: V \rightarrow W\)We defineNuclear/zero space

\[\operatorname{ker}(\Phi):=\Phi^{-1}\left(\mathbf{0}_W\right)=\left\{\boldsymbol{v} \in V: \Phi(\boldsymbol{v})=\mathbf{0}_W\right\} \]

cap (a poem)Image/value field

\[\operatorname{Im}(\Phi):=\Phi(V)=\{\boldsymbol{w} \in W \mid \exists \boldsymbol{v} \in V: \Phi(\boldsymbol{v})=\boldsymbol{w}\} \]

We also call respectively\(V\) cap (a poem)\(W\) because of\(\Phi\) The domain of definition and the domain of value of

Intuitively, the nucleus is\(\Phi\) map to\(W\) Neutral elements in\(\mathbf{0}_W \in W\) vector set of a function (math.)\(\boldsymbol{v} \in V\). It's like being able to pass\(\Phi\) through (a gap)\(V\) Any vector that "arrives" at any of the vectors in the\(\boldsymbol{w} \in W\) The set of A schematic is given in Figure 2.12.

note. Consider a linear mapping\(\Phi: V \rightarrow W\)which\(V, W\) is a vector space.

\(\Phi\left(\mathbf{0}_V\right)=\mathbf{0}_W\) always holds, so\(\mathbf{0}_V \in \operatorname{ker}(\Phi)\). In particular, zero space is never empty.
\(\operatorname{Im}(\Phi) \subseteq W\) be\(W\) A subspace of the\(\operatorname{ker}(\Phi) \subseteq V\) be\(V\) of a subspace.
if and only if\(\operatorname{ker}(\Phi)=\{\mathbf{0}\}\) when\(\Phi\) is a one-shot (injective) (one-to-one correspondence).

marginal notes(zero space and column space). We consider\(\boldsymbol{A} \in \mathbb{R}^{m \times n}\) and a linear mapping\(\Phi: \mathbb{R}^n \rightarrow \mathbb{R}^m, \boldsymbol{x} \mapsto \boldsymbol{A x}\)。

insofar as\(\boldsymbol{A}=\left[\boldsymbol{a}_1, \ldots, \boldsymbol{a}_n\right]\)which\(\boldsymbol{a}_i\) be\(\boldsymbol{A}\) columns, we get

\[\begin{align*} \operatorname{Im}(\Phi) & =\left\{\boldsymbol{A} \boldsymbol{x}: \boldsymbol{x} \in \mathbb{R}^n\right\}=\left\{\sum_{i=1}^n x_i \boldsymbol{a}_i: x_1, \ldots, x_n \in \mathbb{R}\right\} \tag{2.124a} \\ & =\operatorname{span}\left[\boldsymbol{a}_1, \ldots, \boldsymbol{a}_n\right] \subseteq \mathbb{R}^m \tag{2.124b} \end{align*} \]

i.e., something like\(\boldsymbol{A}\) The tensor space of columns, also known ascolumn space. Thus, the column space (like) is\(\mathbb{R}^m\) is a subspace of the space in which\(m\) It is the "height" of the matrix.

\(\operatorname{rk}(\boldsymbol{A})=\operatorname{dim}(\operatorname{Im}(\Phi))\)。
Nuclear/zero space\(\operatorname{ker}(\Phi)\) is the homogeneous system of linear equations\(\boldsymbol{A x}=\mathbf{0}\) The generalized solution and contains all possible\(\mathbb{R}^n\) linear combinations of the elements in which they produce the\(\mathbf{0} \in \mathbb{R}^m\)。
nucleus is\(\mathbb{R}^n\) is a subspace of the space in which\(n\) is the "width" of the matrix.
The kernel focuses on the relationships between columns, which we can use to determine whether/how to represent a column as a linear combination of other columns.

**Example 2.25 (image and kernel of a linear map)**

map (math.)

\[\begin{align*} \Phi: \mathbb{R}^4 \rightarrow \mathbb{R}^2, \quad\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right] & \mapsto\left[\begin{array}{cccc} 1 & 2 & -1 & 0 \\ 1 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right]=\left[\begin{array}{c} x_1+2 x_2-x_3 \\ x_1+x_4 \end{array}\right] \tag{2.125a} \\ & =x_1\left[\begin{array}{l} 1 \\ 1 \end{array}\right]+x_2\left[\begin{array}{l} 2 \\ 0 \end{array}\right]+x_3\left[\begin{array}{c} -1 \\ 0 \end{array}\right]+x_4\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \tag{2.125b} \end{align*} \]

is linear. In order to determine the\(\operatorname{Im}(\Phi)\), we can take the tensor space of the columns of the transformation matrix and get

\[\operatorname{Im}(\Phi)=\operatorname{span}\left[\left[\begin{array}{l} 1 \\ 1 \end{array}\right],\left[\begin{array}{l} 2 \\ 0 \end{array}\right],\left[\begin{array}{c} -1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\right] \tag{2.126} \]

In order to calculate\(\Phi\) of the kernel (zero space), we need to solve the\(\boldsymbol{A} \boldsymbol{x}=\mathbf{0}\), i.e., a system of chi-square equations needs to be solved. For this purpose, we use the Gaussian elimination method to transform the\(\boldsymbol{A}\) Convert to simplified rows in simplest form:

\[\left[\begin{array}{cccc} 1 & 2 & -1 & 0 \\ 1 & 0 & 0 & 1 \end{array}\right] \rightsquigarrow \cdots \rightsquigarrow\left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & -\frac{1}{2} & -\frac{1}{2} \end{array}\right] . \tag{2.127} \]

This matrix is in simplified row-minimal form, and we can compute a basis for the kernel using Minus 1 Trick (see Section 2.3.3). Alternatively, we can represent the non-primitive columns (columns 3 and 4) as linear combinations of the primitive columns (columns 1 and 2). The third column\(\boldsymbol{a}_3\) be tantamount to\(-\frac{1}{2}\) Second column of times\(\boldsymbol{a}_2\). Therefore.\(\mathbf{0}=\boldsymbol{a}_3+\frac{1}{2} \boldsymbol{a}_2\). Similarly, we see that\(\boldsymbol{a}_4=\boldsymbol{a}_1-\frac{1}{2} \boldsymbol{a}_2\)Therefore\(\mathbf{0}=\boldsymbol{a}_1-\frac{1}{2} \boldsymbol{a}_2-\boldsymbol{a}_4\). In summary, this gives the kernel (zero space) as the

\[\operatorname{ker}(\Phi)=\operatorname{span}[\left[\begin{array}{l} 0 \\ \frac{1}{2} \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{c} -1 \\ \frac{1}{2} \\ 0 \\ 1 \end{array}\right]] \tag{2.128} \]

Theorem 2.24(rank-zeroing degree theorem). Forvector space \(V\) cap (a poem)\(W\) as well aslinear map (math.) \(\Phi: V \rightarrow W\)Yes

\[\operatorname{dim}(\operatorname{ker}(\Phi))+\operatorname{dim}(\operatorname{Im}(\Phi))=\operatorname{dim}(V) . \tag{2.129} \]

The rank-zeroing degree theorem is also known as the Fundamental Theorem for linear mappings (Axler, 2015, Theorem 3.22). The following is a direct corollary of Theorem 2.24:

in the event that\(\operatorname{dim}(\operatorname{Im}(\Phi)) < \operatorname{dim}(V)\)follow\(\operatorname{ker}(\Phi)\) is nontrivial, i.e., the kernel contains nothing but\(\mathbf{0}_V\) elements other than the\(\operatorname{dim}(\operatorname{ker}(\Phi)) \geqslant 1\)。
in the event that\(\boldsymbol{A}_{\Phi}\) is relative to some ordered group\(\Phi\) of the transformation matrix and\(\operatorname{dim}(\operatorname{Im}(\Phi)) < \operatorname{dim}(V)\), then the system of linear equations\(\boldsymbol{A}_{\Phi} \boldsymbol{x} = \mathbf{0}\) There are an infinite number of solutions.
in the event that\(\operatorname{dim}(V) = \operatorname{dim}(W)\), then the following three are equivalent:

\[\Phi \text{ It's a one-shot. } \Longleftrightarrow \Phi \text{ It's a full shot. } \Longleftrightarrow \Phi \text{ It's a double shot. } \]

on account of\(\operatorname{Im}(\Phi) \subseteq W\)。