Chapter 1.B Definition of Vector Space 向量空间的定义
这一节主要讨论的是向量空间及其相关性质的定义。
1.18 Definition addition, scalar multiplication 加法，数乘的定义
- An addition on set V is a function that assigns an element u+v ∈ V to each pair of elements u,v ∈ V .
- A scalar multiplication on a set V is a function that assigns an elements λv ∈ V to each λ ∈ F and each v ∈ V .
要定义向量空间，我们自然想到在之前对Fⁿ中加法和数乘的定义，这里我们同样可以这样做，在这里对集合V中的加法和数乘进行定义之后就可以正式的定义向量空间了。
1.19 Definition vector space 向量空间的定义
A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold:
commutativity
u+v = v+u for all u,v ∈ V
associativity(u+v)+w = u+(v+w) and (ab)v = a(bv) for all u,v,w ∈ V and all a,b∈ F
additive identitiesthere exists an element 0 ∈ V such that v+0 = v for all v ∈ V ;
additive inversefor every v ∈ V , there exists w ∈ V such that u+v = 0;
multiplicative identity1v = v for all v ∈ V ;
distributive propertya(u+v) = au+av and (a+b)v = av+bv for all a,b ∈ F and all u,v ∈ V ;
这里的定义与上一节对复数运算的定义很相似，类比理解即可，只需注意这里V中的元素是向量。
1.20 Definition vector,point 向量，点的定义
Elements of a vector space are called vectors or points .
至此，我们可以说V是在F上的一个向量空间，因为V中的数乘是依赖F中的性质的，同理我们可以说Rⁿ和Cⁿ都是在R和C上的向量空间。
1.21 Definition real vector space, complex vector space 实数向量空间，复数向量空间的定义
- A vector space over R is called a real vector space .
- A vector space over C is called a complex vector space .
eg1.22
1.23 Notation Fˢ 符号Fˢ
- If S is a set, then Fˢ denotes the set of functions from S to F.
- For f,g ∈ Fˢ , the sum f+g ∈ Fˢ is the function defined by
(f+g)(x) = f(x)+g(x)
for all x ∈ S .- For λ ∈ F and f ∈ Fˢ, the product λf ∈ Fˢ is the function defined by
(λf)(x) = λf(x)
for all x ∈ S .
这里有点抽象的是Fˢ中的元素是向量也是函数，因为S中的元素是自变量x。这里定义了Fˢ后又定义了Fˢ中的加法和数乘。书上举了一个例子，如果S是间隔[0,1]且F=R，那么R[0,1]就是在间隔[0，1]上所有实值函数的集合，eg1.24。
接下来是引出向量空间一些基本的性质。
1.25 Unique additive identity 唯一加法元
A vector space has a unique additive identity.
1.26 Unique additive inverse 唯一加法逆元
Every element in a vector space has a unique additive inverse.
这两条性质的证明只需用之前定义向量空间的一些性质即可证明。
1.27 Notation -v, w-v 符号-v, w-v
Let v, w ∈ V . Then
- -v denotes the additive inverse of v;
- w-v is defined to be w+(-v).
1.28 Notation V 符号V
For the rest of the book, V denotes a vector space over F.
1.29 The number 0 times a vector 数字0乘以一个向量
0v = 0 for every v ∈ V .
这里等号左边的0是标量，后边的0是向量，证明只需零0v=(0+0)v=0v+0v然后左右同时加上加法逆元即可。
1.30 A number times the vector 0 一个数字乘以一个向量0
a0 = 0 for every a ∈ F.
1.31 The number -1 times a vector 数字-1乘以一个向量
(-1)v = -v for every v ∈ V
1.30，1.31和1.29同理。