本文为《Linear algebra and its applications》的读书笔记

Vector spaces

axiom：公理

公理 1~5 说明 $<V,+>$<V,+> 为一个Abel 群 ($+$+ 为向量的加法运算)

$<R,\oplus,\odot>$<R,⊕,⊙> 为域 ($R$R 为实数集，$\oplus,\odot$⊕,⊙ 为实数集上的加法和乘法运算)，现在在 $R$R 和 $V$V 之间定义标量乘法 $\cdot$⋅

公理 6~10 说明标量乘法 $\cdot$⋅ 对向量加法 $+$+ 满足分配律，标量乘法 $\cdot$⋅ 对域加法 $\oplus$⊕ 满足分配律，标量乘法 $\cdot$⋅ 一致于标量的域乘法 $\odot$⊙，标量乘法 $\cdot$⋅ 有幺元:1, 这里 1 是指域 $R$R 的乘法幺元

One can show that the zero vector is unique, and the vector $-\boldsymbol u$−u, called the negative of $\boldsymbol u$u, is unique for each $\boldsymbol u$u in $V$V . (幺元逆元均唯一)

Technically, $V$V is a real vector space(实数向量空间). All of the theory in this chapter also holds for a complex vector space(复数向量空间) . We will look at this briefly in Chapter 5. Until then, all scalars areassumed to be real.

PROOF
(1). $\because (0+0)\boldsymbol u=0\boldsymbol u+0\boldsymbol u,(0+0)\boldsymbol u=0\boldsymbol u\\ \therefore 0\boldsymbol u+0\boldsymbol u=0\boldsymbol u\\\therefore 0\boldsymbol u=\boldsymbol 0(可约律)$∵(0+0)u=0u+0u,(0+0)u=0u∴0u+0u=0u∴0u=0(可约律)
(2). $\because c(\boldsymbol 0+\boldsymbol 0)=c\boldsymbol 0,c(\boldsymbol 0+\boldsymbol 0)=c\boldsymbol0+c\boldsymbol 0\\ \therefore c\boldsymbol 0+c\boldsymbol 0=c\boldsymbol 0\\\therefore c\boldsymbol 0=\boldsymbol 0(可约律)$∵c(0+0)=c0,c(0+0)=c0+c0∴c0+c0=c0∴c0=0(可约律)
(3). $\because \boldsymbol u+(-1)\boldsymbol u=(1+(-1))\boldsymbol u=0\boldsymbol u=\boldsymbol0 \\\therefore -\boldsymbol u=(-1)\boldsymbol u$∵u+(−1)u=(1+(−1))u=0u=0∴−u=(−1)u

EXAMPLE 1
The spaces $\mathbb R^n$Rn, where $n \geq 1$n≥1, are the premier(首要的) examples of vector spaces. The geometric intuition developed for $\mathbb R^3$R3 will help you understand and visualize many concepts throughout the chapter.

EXAMPLE 2
Let $V$V be the set of all arrows (directed line segments) in threedimensional space, with two arrows regarded as equal if they have the same length and point in the same direction. Define addition by the parallelogram rule, and for each $\boldsymbol v$v in $V$V , define $c\boldsymbol v$cv to be the arrow whose length is $|c|$∣c∣ times the length of $\boldsymbol v$v, pointing in the same direction as $\boldsymbol v$v if $c \geq 0$c≥0 and otherwise pointing in the opposite direction. (See Figure 1.) Show that $V$V is a vector space. This space is a common model in physical problems for various forces.

SOLUTION
The definition of $V$V is geometric, using concepts of length and direction. No $xyz$xyz-coordinate system is involved. An arrow of zero length is a single point and represents the zero vector. The negative of $\boldsymbol v$v is $(-1)\boldsymbol v$(−1)v. So Axioms 1, 4, 5, 6, and 10 are evident. The rest are verified by geometry. For instance, see Figures 2 and 3.

EXAMPLE 3
Let $\mathbb S$S be the space of all doubly infinite sequences of numbers(数的双向无穷序列空间):

If $\{z_k\}${zk​} is another element of $\mathbb S$S, then the sum $\{y_k\}+\{z_k\}${yk​}+{zk​} is the sequence $\{y_k+z_k\}${yk​+zk​} formed by adding corresponding terms of $\{y_k\}${yk​} and $\{z_k\}${zk​}. The scalar multiple $c\{y_k\}$c{yk​} is the sequence $\{cy_k\}${cyk​}. The vector space axioms are verified in the same way as for $\mathbb R^n$Rn.

Elements of $\mathbb S$S arise in engineering, for example, whenever a signal is measured (or sampled) at discrete times. For convenience, we will call $\mathbb S$S the space of (discretetime) signals(信号空间). A signal may be visualized by a graph as in Figure 4.

EXAMPLE 4
For $n \geq 0$n≥0, the set $\mathbb P^n$Pn of polynomials of degree(多项式的次数) at most $n$n consists of all polynomials of the form

If $\boldsymbol p(t)= a_0 \neq 0$p(t)=a0​​=0, the degree of $\boldsymbol p$p is zero. If all the coefficients are zero, $\boldsymbol p$p is called the $zero$zero $polynomial$polynomial. The zero polynomial is included in $\mathbb P^n$Pn.

Analogous to EXAMPLE 3, $\mathbb P^n$Pn is a vecot space. The vector spaces $\mathbb P^n$Pn for various $n$n are used, for instance, in statistical trend analysis of data, discussed in Section 6.8.

EXAMPLE 5
Let $V$V be the set of all real-valued functions(实值函数) defined on a set $\mathbb D$D. Functions are added in the usual way: $\boldsymbol f + \boldsymbol g$f+g is the function whose value at $t$t in the domain $\mathbb D$D is $\boldsymbol f(t) + \boldsymbol g(t)$f(t)+g(t). Likewise, for a scalar $c$c and an $\boldsymbol f$f in $V$V , the scalar multiple $c\boldsymbol f$cf is the function whose value at $t$t is $c\boldsymbol f(t)$cf(t). Two functions in $V$V are equal if and only if their values are equal for every $t$t in $\mathbb D$D. Hence the zero vector in $V$V is the function that is identically zero,$\boldsymbol f(t)=0$f(t)=0 for all $t$t , and the negative of $\boldsymbol f$f is $(-1)\boldsymbol f$(−1)f. Axioms 1 and 6 are obviously true, and the other axioms follow from properties of the real numbers, so $V$V is a vector space.

Subspaces

In many problems, a vector space consists of an appropriate subset of vectors from some larger vector space. In this case, only three of the ten vector space axioms need to be checked; the rest are automatically satisfied.

Some texts replace property (a) in this definition by the assumption that $H$H is nonempty. Then (a) could be deduced from © and the fact that $0\boldsymbol u= \boldsymbol 0$0u=0. But the best way to test for a subspace is to look first for the zero vector. If $\boldsymbol 0$0 is not in $H$H, then $H$H cannot be a subspace and the other properties need not be checked.

Properties (a), (b), and (c） guarantee that a subspace $H$H of $V$V is itself a vector space, under the vector space operations already defined in $V$V . To verify this, note that properties (a), (b), and (c） are Axioms 1, 4, and 6. Axioms 2, 3, and 7–10 are automatically true in $H$H because they apply to all elements of $V$V. Axiom 5 is also true in $H$H, because if $\boldsymbol u$u is in $H$H, then $(-1)\boldsymbol u=-\boldsymbol u$(−1)u=−u is in $H$H by property (c.).

子空间就类似于向量空间的子代数

So every subspace is a vector space. Conversely, every vector space is a subspace (of itself and possibly of other larger spaces). The set consisting of only the zero vector in a vector space $V$V is a subspace of $V$V , called the zero subspace and written as $\{\boldsymbol 0\}${0}.

要证明某个集合是向量空间，也可以证明它是某个向量空间的子空间

EXAMPLE 8
The set

is a subset of $\mathbb R^3$R3 that “looks” and “acts” like $\mathbb R^2$R2, although it is logically distinct from $\mathbb R^2$R2. See Figure 7.

EXAMPLE 9
A plane in $\mathbb R^3$R3 not through the origin is not a subspace of $\mathbb R^3$R3, because the plane does not contain the zero vector of $\mathbb R^3$R3. Similarly, a line in $\mathbb R^2$R2 not through the origin, such as in Figure 8, is not a subspace of $\mathbb R^2$R2.

A Subspace Spanned by a Set 由一个集合生成的子空间

The next example illustrates one of the most common ways of describing a subspace. As in Chapter 1, the term linear combination refers to any sum of scalar multiples of vectors, and $Span\{\boldsymbol v_1,..., \boldsymbol v_p\}$Span{v1​,...,vp​} denotes the set of all vectors that can be written as linear combinations of $\boldsymbol v_1,..., \boldsymbol v_p$v1​,...,vp​.

EXAMPLE 10
Given $\boldsymbol v_1$v1​ and $\boldsymbol v_2$v2​ in a vector space $V$V , let $H = Span\{\boldsymbol v_1,..., \boldsymbol v_p\}$H=Span{v1​,...,vp​}. It is obvious that $H$H is a subspace of $V$V.

In Section 4.5, you will see that every nonzero subspace of $\mathbb R^3$R3, other than $\mathbb R^3$R3 itself, is either $Span\{\boldsymbol v_1, \boldsymbol v_2\}$Span{v1​,v2​} for some linearly independent $\boldsymbol v_1$v1​ and $\boldsymbol v_2$v2​ or $Span\{\boldsymbol v\}$Span{v} for $\boldsymbol v \neq \boldsymbol 0$v​=0. In the first case, the subspace is a plane through the origin; in the second case, it is a line through the origin. (See Figure 9.) It is helpful to keep these geometric pictures in mind, even for an abstract vector space.

The argument in Example 10 can easily be generalized to prove the following theorem.

We call $Span\{\boldsymbol v_1,..., \boldsymbol v_p\}$Span{v1​,...,vp​} the subspace spanned (or generated) by $\{\boldsymbol v_1,..., \boldsymbol v_p\}${v1​,...,vp​}.

Given any subspace $H$H of $V$V , a spanning (or generating) set(生成/张成集) for $H$H is a set $\{\boldsymbol v_1,..., \boldsymbol v_p\}${v1​,...,vp​} in $H$H such that $H = Span\{\boldsymbol v_1,..., \boldsymbol v_p\}$H=Span{v1​,...,vp​}.

The next example shows how to use Theorem 1.

EXAMPLE 11
Let $H$H be the set of all vectors of the form $(a-3b,b-a,a,b)$(a−3b,b−a,a,b), where $a$a and $b$b are arbitrary scalars. That is, let $H = \{(a-3b,b-a,a,b): a\ and\ b\ in\ \mathbb R\}$H={(a−3b,b−a,a,b):a and b in R}. Show that $H$H is a subspace of $\mathbb R^4$R4.
SOLUTION
Write the vectors in $H$H as column vectors. Then an arbitrary vector in $H$H has the form

This calculation shows that $H = Span \{\boldsymbol v_1, \boldsymbol v_2\}$H=Span{v1​,v2​}, where $\boldsymbol v_1$v1​ and $\boldsymbol v_2$v2​ are the vectors indicated above. Thus $H$H is a subspace of $\mathbb R^4$R4.

Example 11 illustrates a useful technique of expressing a subspace $H$H as the set of linear combinations of some small collection of vectors. If $H = Span \{\boldsymbol v_1, ...,\boldsymbol v_p\}$H=Span{v1​,...,vp​}, we can think of the vectors $\boldsymbol v_1, ...,\boldsymbol v_p$v1​,...,vp​ in the spanning set as “handles”(柄) that allow us to hold on to the subspace $H$H. Calculations with the infinitely many vectors in $H$H are often reduced to operations with the finite number of vectors in the spanning set.