3.2 Vector spaces

中文

A vector space $(V,\mathbb{K})$ consists of an additive abelian group $V$ whose elements $u$, $v$, … are called vectors together with a field $\mathbb{K}$ whose elements are termed scalars. The law of composition $u+v$ defining the abelian group is called vector addition. There is also an operation $\mathbb{K}\times V\to V$ called scalar multiplication, which assigns a vector $au\in V$ to any pair $a\in \mathbb{K},u\in V$ . The identity element 0 for vector addition, satisfying $0+u=u$ for all vectors $u$, is termed the zero vector, and the inverse of any vector $u$ is denoted $-u$. In principle there can be a minor confusion in the use of the same symbol $+$ for vector addition and scalar addition, and the same symbol 0 both for the zero vector and the zero scalar. It should, however, always be clear from the context which is being used. A similar remark applies to scalar multiplication $au$ and field multiplication of scalars $ab.$ . The full list of axioms to be satisfied by a vector space is:

(VS1) For all $u,v,w\in V$ and $a,b,c\in \mathbb{K}.$

$$\begin{array}{ccc}u+(v+w)=(u+v)+w& a+(b+c)=(a+b)+c& a(bc)=(ab)c\\ u+v=v+u& a+b=b+a& ab=ba\\ u+0=0+u=u& a+0=0+a=a& a1=1a=a\\ u+(-u)=0;& a+(-a)=0;& a(b+c)=ab+ac.\end{array}$$

(VS2) $a(u+v)=au+av$.

(VS3) $(a+b)u=au+bu$.

(VS4) $a(bv)=(ab)v$.

(VS5) $1v=v$.

A vector space $(V,\mathbb{K})$ is often referred to as a vector space $V$ over a field $\mathbb{K}$ or simply a vector space $V$ when the field of scalars is implied by some introductory phrase such as ‘let $V$ be a real vector space’, or ‘$V$ is a complex vector space’.

Since $v=(1+0)v=v+0v$ it follows that $0v=0$ for any vector $v\in V$ . Furthermore, $(-1)v$ is the additive inverse of $v$ since, by $(VS3),(-1)v+v=(-1+1)v=0v=0.$ . It is also common to write $u-v$ in place of $u+(-v)$, so that $u-u=0$ . Vectors are often given distinctive notations such as $u$, $v$, … or $\vec{u},\vec{v},\dots .$ , etc. to distinguish them from scalars, but we will only adopt such notations in specific instances.

Example 3.5

The set ${\mathbb{K}}^n$ of all $n$-tuples $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ where $x_i\in \mathbb{K}$ is a vector space, with vector addition and scalar multiplication defined by

$$\begin{array}{c}\mathbf{x}+\mathbf{y}=(x_1+y_1,x_2+y_2,\dots ,x_n+y_n),\\ a\mathbf{x}=(a{x}_1,a{x}_2,\dots ,a{x}_n).\end{array}$$

Specific instances are ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ . Sometimes the vectors of ${\mathbb{K}}^n$ will be represented by $n\times 1$ column matrices and there are some advantages in denoting the components by superscripts,

$$\mathbf{x}=\left(\begin{array}{c}{x}^1\\ x^2\\ ⋮\\ x^n\end{array}\right).$$

Scalar multiplication and addition of vectors is then

$$a\mathbf{x}=\left(\begin{array}{c}a{x}^1\\ a{x}^2\\ ⋮\\ a{x}^n\end{array}\right),\mathbf{x}+\mathbf{y}=\left(\begin{array}{c}{x}^1+y^1\\ x^2+y^2\\ ⋮\\ x^n+y^n\end{array}\right).$$

Exercise

Verify that all axioms of a vector space are satisfied by ${\mathbb{K}}^n$

Example 3.6

Let ${\mathbb{K}}^{\infty }$ denote the set of all sequences $u=(u_1,u_2,u_3,...)$ where $u_i\in \mathbb{K}$. This is a vector space if vector addition and scalar multiplication are defined as in Example 3.5:

$$\begin{array}{c}u+v=(u_1+v_1,u_2+v_2,u_3+v_3,\dots ),\\ au=(a{u}_1,a{u}_2,a{u}_3,\dots ).\end{array}$$

Example 3.7

The set of all $m\times n$ matrices over the field $\mathbb{K}$, denoted $M^{(m,n)}(\mathbb{K})$ , is a vector space. In this case vectors are denoted by $\mathsf{A}=[a_{ij}]$ where $i=1,\dots ,m,j=1,\dots ,n$ and $a_{ij}\in \mathbb{K}$ . Addition and scalar multiplication are defined by:

$$\mathsf{A}+\mathsf{B}=[a_{ij}+b_{ij}],c\mathsf{A}=[c{a}_{ij}].$$

Although it may seem a little strange to think of a matrix as a ‘vector’, this example is essentially no different from Example 3.5, except that the sequence of numbers from the field $\mathbb{K}$ is arranged in a rectangular array rather than a row or column.

Example 3.8

Real-valued functions on ${\mathbb{R}}^n$ , denoted $\mathcal{F}({\mathbb{R}}^n)$ , form a vector space over $R$. As described in Section 1.4, the vectors in this case can be thought of as functions of $n$ arguments,

$$f(\mathbf{x})=f(x_1,x_2,\dots ,x_n),$$

and vector addition $f+g$ and scalar multiplication $af$ are defined in the obvious way,

$$(f+g)(\mathbf{x})=f(\mathbf{x})+g(\mathbf{x}),(af)(\mathbf{x})=af(\mathbf{x}).$$

The verification of the axioms of a vector space is a straightforward exercise.

More generally, if $S$ is an arbitrary set, then the set $\mathcal{F}(S,\mathbb{K})$ of all $\mathbb{K}$-valued functions on $S$ forms a vector space over $\mathbb{K}$. For example, the set of complex-valued functions on $R^n$, denoted $\mathcal{F}({\mathbb{R}}^n,\mathbb{C})$ , is a complex vector space. We usually denote $\mathcal{F}(S,\mathbb{R})$ simply by $\mathcal{F}(S)$ , taking the real numbers as the default field. If $S$ is a finite set $S=\{1,2,\dots ,n\}$ , then $\mathcal{F}(S,\mathbb{K})$ is equivalent to the vector space ${\mathbb{K}}^n$ , setting $u_i=u(i)$ for any $u\in \mathcal{F}(S,\mathbb{K})$

When the vectors can be uniquely specified by a finite number of scalars from the field $\mathbb{K}$, as in Examples 3.5 and 3.7, the vector space is said to befinite dimensional. The number of independent components needed to specify an arbitrary vector is called the dimension of the space; e.g., ${\mathbb{K}}^n$ has dimension $n$, while $M^{m,n}(\mathbb{K})$ is ofdimension $mn$. On the other hand, in Examples 3.6 and 3.8 it is clearly impossible to specify the vectors by a finite number of scalars and these vector spaces are said to be infinite dimensional. A rigorous definition of these terms will be given in Section 3.5.

Example 3.9

A set $M$ is called a module over a ring $R$ ifit satisfies all the axioms (VS1)– (VS5) with $R$ replacing the field $\mathbb{K}$. Axiom (VS5) is only included ifthe ring has an identity. This concept is particularly useful when $R$ is a ring of real or complex-valued functions on a set $S$ such as the rings $\mathcal{F}(S)$ or $\mathcal{F}(S,\mathbb{C})$ in Example 3.8.

A typical example of a module is the following. Let $\mathcal{C}({\mathbb{R}}^n)$ ) be the ring of continuous real-valued functions on ${\mathbb{R}}^n$ , sometimes called scalar fields, and let ${\mathcal{V}}^n$ be the set of al $n$-tuples ofreal-valued continuous functions on ${\mathbb{R}}^n$ . A typical element of ${\mathcal{V}}^n$ , called a vector field on ${\mathbb{R}}^n$ , can be written

$$\mathbf{v}(\mathbf{x})=(v_1(\mathbf{x}),v_2(\mathbf{x}),\dots ,v_n(\mathbf{x}))$$

where each $v_i(\mathbf{x})$ is a continuous real-valued function on ${\mathbb{R}}^n$ . Vector fields can be added in the usual way and multiplied by scalar fields,

$$\begin{array}{c}\mathbf{u}+\mathbf{v}=(u_1(\mathbf{x})+v_1(\mathbf{x}),\dots ,u_n(\mathbf{x})+v_n(\mathbf{x})),\\ f(\mathbf{x})\mathbf{v}(\mathbf{x})=(f(\mathbf{x})v_1(\mathbf{x}),f(\mathbf{x})v_2(\mathbf{x}),\dots ,f(\mathbf{x})v_n(\mathbf{x})).\end{array}$$

The axioms (VS1)–(VS5) are easily verified, showing that ${\mathcal{V}}^n$ is a module over $\mathcal{C}({\mathbb{R}}^n)$ This module is finite dimensional in the sense that only a finite number ofcomponent scalar fields are needed to specify any vector field. Of course ${\mathcal{V}}^n$ also has the structure of a vector space over the field $R$, similar to the vector space $\mathcal{F}({\mathbb{R}}^n)$ ) in Example 3.8, but as a vector space it is infinite dimensional.