Chapter 1.A Rⁿ and Cⁿ
本节主要定义了复数的概念和一些运算性质，以及列表list和 F 及 Fⁿ 等概念。
Complex Numbers 复数
1.1 Definition complex numbers 复数的定义
- A complex number is an ordered pair (a,b), where a,b ∈ R , but we will write this as a+bi.
- The set of all complex numbers is denoted by C :
C = {a+bi:a,b ∈ R } - Addition and multiplication on C are defined by
(a+bi)+(c+di) = (a+c)+(b+d)i,
(a+bi)(c+di) = (ac-bd)+(ad+bc)i;
here a,b,c,d ∈ R .
eg1.2
1.3 Properties of complex arithmetic 复数运算的一些性质
commutativity
α+β = β+α and αβ = βα for all α,β∈ C;
associativity(α+β)+λ = α+(β+λ) and (αβ)λ = α(βλ) for all α,β,λ∈ C;
identitiesλ + 0 = λ and λ1 = λ for all λ∈ C;
additive inversefor every α ∈ C , there exists a unique β ∈ C such that α+β = 0;
multiplicative inversefor every α ∈ C with α ≠ 0, there exists a unique β ∈ C such that αβ = 1;
distributive propertyλ(α+β) = λα+λβ for all λ,α,β ∈ C.
这里的一些性质和实数的性质非常相似，想要证明可以用实数的一些性质和之前对复数的定义来证明，eg1.4。
Definition -α,subtraction,1/α,division 加法逆元-α，减法，乘法逆元1/α，除法的定义
Let α,β ∈ C .
- Let -α denote the additive inverse of α. Thus -α is the unique complex number such that
α+(-α) = 0. - Subtraction on C is defined by
β-α = β+(-α). - For α ≠ 0, let 1/α denote the multiplicative inverse of α. Thus 1/α is the unique complex number such that
α(1/α) = 1. - Division on C is defined by
β/α = β(1/α).
此处同上。
1.6 Notation F 符号F
Throughout this book, F stands for either R or C.
值得注意的是，这里F中的每个元素都是标量，与之相对的是向量，书上还定义了a次方，但这里就不赘述了，eg1,7。
Lists 列表
1.8 Definition list,length 列表和长度的定义
Suppose n is a nonnegative integer. A list of length n is an ordered collection of n elements (which might be numbers, other lists, or more abstract entities) separated by commas and surrounded by parentheses. A list of length n looks like that:
(x₁,...,xₙ).
Two lists are equal if and only if they have the same length and the same elements in the same order.
这里要强调的有两点，一是列表中的元素都是非负整数，二是列表是有长度的。列表和集合的不同之处在于前者内部的元素顺序和重复是有意义的，而后者没有，ex1.9。
Fⁿ
Definition Fⁿ Fⁿ的定义
Fⁿ is the set of all lists of length n of elements of F:
Fⁿ = {(x₁,...,xₙ):xⱼ ∈ F for j = }.
For (x₁,...,xₙ) ∈ Fⁿ and j ∈ {1,...,n}, we say that xⱼ is the jᵗʰ coordinate of (x₁,...,xₙ).
eg1.11
1.12 Definition addition in Fⁿ Fⁿ中加法的定义
Addition in Fⁿ is defined by adding corresponding coordinates:
(x₁,...,xₙ)+(y₁,...,yₙ) = (x₁+y₁,...,xₙ+yₙ)
1.13 Commutativity of addition in Fⁿ Fⁿ中加法交换律的定义
If x,y ∈ Fⁿ, then x+y = y+x.
此处证明用上面定义与F中的交换律即可。
1.14 Definition 0 0的定义
Let 0 denote the list of length n whose coordinates are all 0:
0 = (0,...,0).
这里用0来定义确定长度中所有元素为0的列表，要区分式子中的0是数字还是列表主要看其他项是属于什么域，eg1.15。
1.16 Definition additive inverse in Fⁿ Fⁿ中加法逆元的定义
For x ∈ Fⁿ, the additive inverse of x, denoted -x, is the vector -x ∈ Fⁿ such that
x+(-x) = 0.
In other words, if x = (x₁,...,xₙ), then -x = (-x₁,...,-xₙ).
1.17 Definition scalar multiplication in Fⁿ Fⁿ中数乘的定义
The product of a number λ and a vector in Fⁿ is computed by multiplying each coordinate of the vector by λ:
λ(x₁,...,xₙ) = (λx₁,...,λxₙ);
here λ ∈ F and (x₁,...,xₙ) ∈ Fⁿ.
这里我们定义了Fⁿ中加法和数乘的计算，这样让我们便于后面的计算，至于向量和向量的乘法并不能简单的照搬F中的定义，所以这里暂按不表。