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Deepening the concept of linear algebra [23] 01 - points coordinate points and vectors vectors

2022-07-28 01:12:00 【Franklin】

Preface ：

In this column article 7,12 We all mentioned the concept of vector , The expression of vectors in the two articles starts from the definitions of algebra and geometry , but , I don't think it's deep enough . This section will discuss it in depth from the perspective of algebra and geometry combined with practice .

This section starts from $\large \mathbb{R}^{n}$ Of n D coordinate system 【n coordinates】 set out , Elaborated the fundamental meaning of vector carefully and thoroughly , I hope I can help you to understand .

Summary ：

A vector is a set of logically related data .【 It's that simple , It's over , Understandable PASS 了】

Let me write it out front ：

Our Chinese achievements in the field of mathematics are too few in the twothousand year long river , Until recently , Mr. Zhang Yitang from Pinghu, Zhejiang , A solvable problem between number theory and prime numbers , Promoted the progress of Mathematics , So as to be listed in the hall of fame .

Timeline of Mathematics | Mathigon

1【 Get ready 】 The concept of ：

Arthur Cayley:

1846 year , This guy 25 At the age of , Proposed matrix algebra , Say that the four-dimensional points of a four-dimensional space can be correctly expressed in geometry , Without any supernatural explanation .

thus , The source of multidimensional vector is first related to the expression of points in multidimensional space . This article also follows this idea , Let's discuss it from the point of view of multidimensional space .

1.1 The difference between mathematics and vectors in other fields

Physics ：

In Physics , In particular, gas research pressure and temperature , Vectors include positions 、 Time 、 temperature 、 Pressure . Fluid mechanics research field , The vector represents , Direction of fluid particles , but , These are not vectors in mathematical concepts .

biological ：

Research Sharks and Sardiness When the expression .

mathematics ：

Express the change of things as a function functions, and function In fact, it's a group n A number to m Number conversion ： Expressed mathematically as ： $\large f:\mathbb{R}^{n}\Rightarrow \mathbb{R}^{m}$ , And vector is the group of data that expresses changes .

1.2 The importance of vectors ：

Vector is the basis of the application of the whole linear algebra . Vector is also the constituent element of vector space , It is the basis of multivariable calculus . Vector is the basic element of mathematical multi-dimensional space transformation .

2 The essence of vectors

The essence of a vector is a set of related sequences ：

In the previous analysis , Vectors can be understood as moving , Shiguangyan said “ A row or column of numbers , These numbers are logically related . We call it a vector ”.Essence of linear algebra In the video, the definition of vector starts from two basic base vectors , The basic mathematical and geometric explanations of vectors are given .Dr.Trefor Bazett stay linear Algebra The course mentions , A vector is a set of guidelines （an instruction） It shows the transformation of vectors and the properties of motion .

stay 《vector calculus linear algebarta and differential Forms.》 in , The essential difference between vector and point real number space is given ：

point yes position Location
vector yes increment perhaps displacement

【 case , Examples in the book are from traditional physics , Define the vector in physics as having magnitue Quantity and direction The attribute of direction is vector , Like speed 、 force . however , this magnitue Quantity and direction Two concepts of direction , But these two definitions are too limited . such as , Temperature is only a concept of quantity , however , The temperature can change , For example, it can change from low temperature to high temperature , In this way, there is a warming , Cooling in two directions , therefore , It can also be defined as a vector .】

2.1 Real space Real Space $\huge \mathbb{R}^{n}$ : The difference between points and vectors ,

Before expressing all the concepts , Let's understand the expression of real number space .

$\huge \mathbb{R}^{n}$

This symbol , Just state n Real columns (orderd lists of real number) The space expressed space.

We are most familiar with Cartesian space , Two coordinates x,y To express a plane is $\huge \mathbb{R}^{^{2}}$ ,x,y,z The expression is $\huge \mathbb{R}^{3}$

for example , coordinate x,y yes $\huge \mathbb{R}^{^{2}}$ An element of space .x,y,z yes $\huge \mathbb{R}^{3}$ The elements of space , and , $\huge \mathbb{R}^{n}$ The elements of space , It's a group. n A set of numbers in order , Or we think , The element has x1,x2,...,xn Coordinates of .

Both points and vectors can say this $\huge \mathbb{R}^{n}$ The elements of space , So what's the difference , Fundamentally speaking , The element representing the position is called point , And characterization displacement and increment, I understood as a migrate Relationship or Amount of change When we use vector .

General ：

Use parentheses to express points ,

$\begin{pmatrix} x1\\ ..\\ x2 \end{pmatrix}$

Use square brackets to express vectors ：

$\begin{bmatrix} x1\\ ...\\ x2 \end{bmatrix}$

2.1 Point Point and Cartesina The limitations of Cartesian space ：

from 2.1, Usually we express n The expression of the point of a dimensional vector is his n A coordinate point(x,y,z,....) To express , These coordinates are n A list of real numbers .

for example ： stay $\huge \mathbb{R}^{^{2}}$ In the plane , Represent a （2,3） The point of ,

$\begin{pmatrix} 2\\ 3 \end{pmatrix}$

point Used to express positional The data is better , But it's not good for data representing change or migration .

Now let's look at the representation of vectors ：

Above picture , There are three vectors , They are all the same vector , Of course , their original Dissimilarity

$\begin{bmatrix} 2\\ 3 \end{bmatrix}$

We are in the essence of vectors section , Start , We discussed , vector vector yes increment perhaps displacement, such , We can compare the difference between vectors and coordinate points visually , Coordinate points mainly represent the position in the coordinate system , The vector is composed of the origin and the length of the transformation or migration , A vector number order can represent countless similar changes 】

In this way, even if it is a complex data set like stocks , It can also be regarded as a set of vectors .

2.2 The difference between points and vectors

front , What we are talking about is the location information , therefore , The addition of dots is meaningless , such as , Beijing + Shanghai is meaningless . And vectors express migration and transformation , that , The addition of vectors is meaningful , such as , Distance to Beijing + Distance to Shanghai .

General ：

Use parentheses to express points ,

$x = \begin{pmatrix} x1\\ ..\\ x2 \end{pmatrix}$

Use square brackets to express vectors ：

$\vec{x} = \begin{bmatrix} x1\\ ...\\ x2 \end{bmatrix}$

Yes $\large \mathbb{R}^{n}$ For the elements of , Is an ordered list of numbers , It can be a point or a vector , however , In linear algebra , $\large \mathbb{R}^{n}$ You need to think of its elements as vectors by default . This is also the correct language expression and tool for multidimensional complex number operation .

【 case 】 Of course , In essence , Vectorial increments The study of , The essence is also based on the correlation of points , This point will be gradually understood in the next section .

2.3 Vector addition and subtraction and points

The essence of vectors , In fact, it represents the transformation relationship of points , such as ：

Above picture , vector $\vec{ab}$ In fact, it represents , spot a, point-to-point b Different . stay $\large \mathbb{R}^{3}$ In my space , spot a, point-to-point b Transformation of , from x,y,z From three coordinates , In every coordinate , Unit length has been increased 1（0+1,0+1,0+1）

then , We will have a b light , One more to c(2,2,0) The transformation of points , vector $\large \vec{bc}$ Characterized b Point to c Different points ,（1+1,1+1,1+0）

So a little a, spot b Represented vector $\vec{ab}$ , Sum point b spot c Represented vector $\large \vec{bc}$

that , Naturally a little a, spot c Represented vector $\large \large \vec{ac}$ , So what is their relationship ？

Now? , Let's think about it , If ,a The point changes directly to c How about it? ？ We can also understand a Change to b, Again by b Change to c,

That is to say , $\large \large \large \vec{ac} = \large \large \vec{ab} + \large \large \vec{bc}$

This leads to the concept of vector addition , It's the motion of points or the sum of transformations , And calculation method , It can be obtained by accumulating the component distribution of each coordinate ,

【 case If this transformation is an elementary linear transformation , for example , Addition, subtraction, multiplication, division, etc , Then we call it the linear transformation of vectors .】

If , Make ab=w,ac = u,bc =v, Yes $\large \vec{u} = \large \large \vec{w} + \large \large \vec{v}$

If v1,v2... $\large v_{n}$ Respectively $\huge \mathbb{R}^{n}$ Inside the space , vector v The variation of each coordinate ,

If w1,w2... $\large w_{n}$ Respectively $\huge \mathbb{R}^{n}$ Inside the space , vector w The variation of each coordinate ,

then , We also express these coordinates vertically , Then the general expression of vector addition and subtraction is as follows ：

$\LARGE \vec{u}=\vec{v} + \vec{w} = \begin{bmatrix}v1\\ ...\\ v_{n} \end{bmatrix} + \begin{bmatrix}w1\\ ...\\ w_{n} \end{bmatrix} = \begin{bmatrix}v1+w1\\ ...\\ v_{n}+w_{n} \end{bmatrix}$