4.1 矢量空间和矢量空间标准

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  矢量空间和矢量空间标准

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  Vector spaces are the topic where linear algebra starts to get interesting and leads into pure abstract mathematics. Vector spaces are essentially mathematical objects where you can study operations and how a collection of objects works together.
  ::矢量空间是线性代数开始变得有趣并引向纯抽象数学的主题。矢量空间本质上是数学对象,可以在那里研究操作和集合物体如何一起工作。

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  Though you may have not known it, we've been working in vector spaces since the beginning of the course. Vector spaces help us generalize mathematics and can find ways to find general solutions to simple problems. However, for you all of you all who  are not as much fans of theoretical math we have many different ways to apply vector spaces to the real world. For example, infinite dimensional vector spaces have many different applications in engineering and theoretical physics.
  ::尽管你可能还不知道这一点,但我们从课程开始就一直在矢量空间中工作。矢量空间有助于我们普及数学,并找到解决简单问题的一般方法。然而,对于你们这些没有那么多理论数学粉丝的人来说,我们有许多不同的方法将矢量空间应用到现实世界。例如,无限的维度矢量空间在工程和理论物理方面有许多不同的应用。

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  Now, let's formally define what a vector space is.
  ::现在,让我们正式定义矢量空间是什么。

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  Definition:  Vector Space
  ::定义:矢量空间

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  - A vector space is a nonempty set of mathematical objects with the binary operations of addition and scalar multiplication satisfying the following axioms:
  ::- 矢量空间是一组非空的数学天体,其附加和计算乘法的二进制操作满足下列等离子:

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  1. $\begin{align*}\text{If } \vec{u} \in V, \vec{v} \in V \to \vec{u} + \vec{v} \in V\end{align*}$  where (+) denotes the some of the two vectors
     ::如果 uV,vVVvV,则(+) 表示两个矢量中的某些矢量。
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  2. $\begin{align*}\vec{u} + \vec{v} = \vec{v} + \vec{u}\end{align*}$  
     ::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
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  3. $\begin{align*}(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})\end{align*}$  
     :u)+w(v)
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  4. $\begin{align*}\exists\vec{v} \in V \text{ s.t. } \forall{\vec{u}} \in \text{V }, \vec{u} + \vec{v} = \vec{u} \text{ where } \vec{v} = \vec{0}\end{align*}$  
     ::Vs. t. V. V. V.
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  5. $\begin{align*}\forall{\vec{v}}\in V \exists \vec{u} \in V \text{ s.t. } \vec{v} + \vec{u} = \vec{0}\text{ and } \vec{u} = -\vec{v}\end{align*}$
     ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}哦! {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}哦! {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}哦! {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}哦!
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  6. For a scalar multiple $\begin{align*}c \text{ and } \vec{v} \in V, c\vec{v} \in V\end{align*}$
     ::以 scalar 倍数c 和 vvvvvvv
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  7. Also,  $\begin{align*}c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}\end{align*}$
     ::还有, c(uv) =cucv
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  8. For another scalar multiple  $\begin{align*}d, (c+d)\vec{u} = c\vec{u} + d\vec{u}\end{align*}$
     ::另一弧度多 d, (c+d) u cu du
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  9. $\begin{align*}c(d\vec{u}) = (cd)\vec{u}\end{align*}$
     ::c(du__)=(cd)u_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  10. $\begin{align*}1\vec{u} = \vec{u}\end{align*}$
      ::一、一、一、一、二、三、三、三、四、四、四、四、四、四、四、五、五、五、五、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、六、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十

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  Now let's prove a couple of theorems and look at a couple of examples about vector spaces.
  ::现在让我们来证明几个理论,看看关于矢量空间的几个例子。

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  One theorem that we can deduce about vector spaces is that  $\begin{align*}\forall{v} \in V, 0\vec{v} = \vec{0}\end{align*}$ .
  ::关于矢量空间,我们可以推断出的一个理论是 vV,Ov0。

   

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  Theorem 1:  $\begin{align*}0\vec{v} = \vec{0}\end{align*}$
  ::定理1: 0v0

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  Proof:
  ::证据:

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    $\begin{align*}0\vec{v}\\ 0 = 0 + 0 \text{ (By the definition of 0 in a vector space)}\\ (0 + 0)\vec{v} = 0\vec{v}\\ 0\vec{v} + 0\vec{v} = 0\vec{v}\\ 0\vec{v} + 0\vec{v} + (-0\vec{v}) = 0\vec{v} + (-0\vec{v})\text{ Applying additive inverses }\\ 0\vec{v} + 0 = 0\\ 0\vec{v} = 0 \text{ From the definition of 0}\end{align*}$
  ::0v_0=0+0 (根据矢量空间 0( 0+0) v0v0v0v0v0v0v0v}0v0v0v00v}( ~0v)=0v( -0v}) 应用 0v( 0v00=00v0) 定义 0

   

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  Theorem 2:  $\begin{align*}c\vec{0} = \vec{0}\end{align*}$
  ::定理2: c00

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  Proof: 
  ::证据:

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  $\begin{align*}\vec{0} + \vec{0} = \vec{0}\text{ By the definition of the zero vector}\\ \text{So... }c(\vec{0}) = c(\vec{0} + \vec{0})\\ c(\vec{0}) = c\vec{0} + c\vec{0}\\ \text{Also } c\vec{0} = \vec{0} + c\vec{0}\text{Again, by the zero vector}\\ \text{Now, we end up getting that }a\vec{0} + a\vec{0} = \vec{0} + a\vec{0}\\ \text{Hence, } a\vec{0} + a\vec{0} + (-a\vec{0}) = \vec{0} + a\vec{0} + (-a\vec{0})\\ a\vec{0} = \vec{0}\\ \end{align*}$
  ::根据零向量的定义... c(0)=c(0)=c(0)=c0(0)=c(0)=c(0)=c(0)=c(0)=c(c)=c(c)=c(0)=c(0)=c(0)=c(0)=c(0)=c(0)=c(0)=c(0)=c(0)=c0(0)=A(0)_(0)_(0)=g(0)=(0)=(0)=c(0)=c(0)=c(0)=c(0)=c(0)=c(0)

   

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  Theorem 3:  $\begin{align*}(-1)\vec{u} = -\vec{u}\end{align*}$
  ::3: (- 1) 定理

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  Proof:
  ::证据:

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  • In order to show this is true we must show that  $\begin{align*}\vec{u} + (-1)\vec{u} = \vec{0}\end{align*}$  because that also is what happens with  $\begin{align*}-\vec{u}\end{align*}$  by the definition of the additive inverse of a vector.
    ::为了证明这是真的,我们必须证明u(-1)u0,因为根据矢量的反添加剂定义,这也是-u发生的情况。

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  • Now, by the distributivity property of vector spaces  $\begin{align*}\vec{u} + (-1)\vec{u} = \vec{u}(1+(-1))\end{align*}$ .
    ::现在,根据矢量空间的分布属性 u(-1)uuu(1+(-1))。

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  • And,  $\begin{align*}\vec{u} + (-1)\vec{u} = 0\vec{u}\text{ which from an earlier theorem gives us } \vec{0}\end{align*}$ .
    ::和,u(-1-1)uOu (-1-IO) 从早期的定理 给了我们0(0)。

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  • Now we have shown that  $\begin{align*}(-1)\vec{u}\end{align*}$  is equivalently the additive inverse of $\begin{align*}\vec{u}\end{align*}$  , or  $\begin{align*}-\vec{u}\end{align*}$  
    ::现在我们已经证明 (-1) 与 u 或- u 的反比等同。

   

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  Theorem 4: From theorem 3 derive that,  $\begin{align*}(-c)\vec{u} = -c\vec{u}\end{align*}$
  ::论理4:从理论3得出,(-c)u

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  • We need to repeat a similar process as we did in the last theorem. 
    ::我们需要重复我们在最后一个理论中所做的类似进程。

   

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  • We must show that because  $\begin{align*}-c\vec{u}\end{align*}$  is the additive inverse of the vector  $\begin{align*}c\vec{u}\end{align*}$  then so is the vector  $\begin{align*}(-c)\vec{u}\end{align*}$  
    ::我们必须证明,因为-cu是矢量 cuc 的反相,所以矢量(-c)u也是反相。

   

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  • Similarly to the last proof,  $\begin{align*}c\vec{u} + (-c)\vec{u} = \vec{u}(c + (-c))\end{align*}$ , by the distributive property of vector spaces.
    ::与最后的证据相似,即矢量空间的分配属性的 cu(-c) uuu(c+(c)-c)。

   

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  • So we get that  $\begin{align*}c\vec{u} + (-c)\vec{u} = 0\vec{u} \to c\vec{u} + (-c)\vec{u} = \vec{0}\end{align*}$
    ::所以,我们得到了这个 (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c) (-c)

   

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  • Hence,  $\begin{align*}(-c)\vec{u}\end{align*}$  is the additive inverse of  $\begin{align*}c\vec{u}\end{align*}$  so  $\begin{align*}(-c)\vec{u} = -c\vec{u}\end{align*}$
    ::因此,(c)u是cuc的反比,所以(c)uucu

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  Theorem 5: The zero vector is unique
  ::定理5: 零向量是独一无二的

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  • For the sake of contradiction assume that the zero vector is not unique and that  $\begin{align*}\vec{0}, \vec{0'}\end{align*}$  
    ::出于矛盾的考虑,假设零向量并非独一无二,而且 0,0,0

   

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  • Hence, by the definition of the zero vector you get that  $\begin{align*}\vec{0} + \vec{0'} = \vec{0}\text{ and also }\\ \vec{0} + \vec{0'} = \vec{0'}\text{ so we get that }\\ \vec{0} = \vec{0'}\text{ so these two vectors are not unique }\end{align*}$  
    ::因此,根据零向量的定义,您可以得到 000000 和 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

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  Theorem 6: Additive inverses are unique
  ::论理6: 补充反演是独一无二的

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  • By definition let  $\begin{align*}\vec{x'} \text{ and } \vec{x''}\end{align*}$  are both the additive inverses of the vector  $\begin{align*}\vec{x}\end{align*}$  
    ::按定义, let x 和 x 既是矢量 x 的添加反正值, 也是矢量 x 的添加反正值 。

   

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  • So,  $\begin{align*}\vec{x'} = \vec{x'} + \vec{0} (\text{ By the additive identity definition})\\ \vec{x'} = \vec{x'} + (\vec{x} + \vec{x''}) (\text{ Because the zero vector is the sum of a vector and its additive identity})\\ \text{By simplifying}\\ \vec{x'} = \vec{x'} + \vec{x} + \vec{x''} \to \text{ (By associativity) } \vec{x'} = (\vec{x'} + \vec{x}) + \vec{x''}\\ \vec{x'} = \vec{x''}\text{ By that also being the additive inverse}\\ \text{Now we have contradicted our statement and shown that the additive identity is unique.}\end{align*}$  
    ::xx0(根据添加剂身份定义)xxx(xxx)(因为零矢量是矢量及其添加剂身份的总和)By 简化xxxxxx(双联性)x(xxx)+xxxxx)(这也正是我们自相矛盾的说法,并表明添加剂身份是独一无二的。

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  Theorem 7: If  $\begin{align*}\vec{u} + \vec{w} = \vec{v} + \vec{w} \to \vec{u} = \vec{v} \end{align*}$
  ::理论7:如果uwvvwuuv

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  Proof:
  ::证据:

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  $\begin{align*}\vec{v} + \vec{w} = \vec{u} + \vec{w}\\ \vec{v} + \vec{w} + (-\vec{w}) = \vec{u} + \vec{w} + (-\vec{w})\\ \vec{v} + (\vec{w} + (-\vec{w})) = 0\vec{w}\\ 0\vec{w} = \vec{0}\\ \vec{u} + (\vec{w} + (-\vec{w})) = 0\vec{w}\\ 0\vec{w} = \vec{0}\\ \vec{v} + \vec{0} = \vec{u} + \vec{0}\\ \vec{v} + \vec{0} = \vec{v}\\ \vec{u} + \vec{0} = \vec{u}\\ \vec{v} = \vec{u}\\ \end{align*}$
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}( - -) {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}(~) {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}(~) {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}(~) {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}(~)

   

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  When we work with vector spaces we work with vectors and scalars. We take our set of scalars in what we call a field.
  ::当我们与矢量空间一起工作时,我们与矢量和卡路里一起工作。我们在我们所称的字段中采取一套卡路里。

   

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  Here, we may have a field which generally is denoted by  $\begin{align*}\mathbb{F}\end{align*}$  and is usually the set of reals or complex numbers. 
  ::在这里,我们可能有一个通常由F表示的领域,通常是一套真实数字或复杂数字。

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  A field is another abstract mathematical  set in which all the operations of addition, subtraction, multiplication and division are defined, loosely speaking. By this we have that $\begin{align*}a,b\in\mathbb{F}, a + b\in\mathbb{F}, a-b\in\mathbb{F}, a\cdot b\in\mathbb{F}\text{ & } a\div b\in \mathbb{F}\\ a - b = a + (-b) \text{ b's additive inverse}\\ a\div b = a\cdot b^{-1} \text{b's multiplicative inverse} \end{align*}$  . So everything has an additive and a multiplicative inverse that is also in the set.
  ::字段是另一个抽象的数学集,其中定义了所有添加、减法、乘法和除法的操作,简洁易懂。因此,我们有一个,bF,a+bF,a-bF,a-bF & abFa-b=a+(b)b的添加剂反向反向。因此,一切都有一个添加剂和多倍反向,也包含在集中。

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  Fields are mostly applicable to theoretical math in abstract algebra and geometry.
  ::字段大多适用于抽象代数和几何学中的理论数学。

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  Lastly, ponder this statement: The integers are not a field, but the rationals are. Try to think of a good explanation why. 
  ::最后,请仔细考虑一下这句话:整数不是一个字段,而是理性。试着想一个好的解释。

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