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Basic Mathematics  基本数学

Contents 目录

1 Basic Skills ..... 2
1 基本技能 ..... 2

1.1 Practice Questions ..... 2
1.1 练习题 ..... 2

2 Linear Algebra ..... 3
2 线性代数 ..... 3

2.1 Matrices and Vectors ..... 3
2.1 矩阵和向量 ..... 3

2.1.1 Definitions ..... 3
2.1.1 定义 ..... 3

2.1.2 Notation ..... 4
2.1.2 符号 ..... 4

2.1.3 Addition ..... 4
2.1.3 加法 ..... 4

2.1.4 Subtraction ..... 5
2.1.4 减法 ..... 5

2.1.5 Multiplication by a scalar . ..... 5
2.1.5 标量乘法。..... 5

2.1.6 Multiplication of two matrices ..... 5
2.1.6 两个矩阵的乘法 ..... 5

2.1.7 Motivation for matrix-matrix multiplication ..... 7
2.1.7 矩阵矩阵乘法的动机 ..... 7

2.1.8 Matrix-vector multiplication ..... 8
2.1.8 矩阵-向量乘法 ..... 8

2.1.9 Special Matrices ..... 8
2.1.9 特殊矩阵 ..... 8

2.1.10 Scalar products and orthogonality ..... 10
2.1.10 标量积和正交性 ..... 10

2.2 Linear Systems ..... 11
2.2 线性系统 ..... 11

2.3 Determinants ..... 12
2.3 行列式 ..... 12

2.3.1 Using determinants to invert a matrix ..... 14
2.3.1 使用行列式来求逆 矩阵..... 14

2.4 Eigenvalues and Eigenvectors ..... 15
2.4 特征值和特征向量 ..... 15

3 Differentiation and Integration ..... 21
3 差异化和整合 ..... 21

3.1 Differentiation ..... 21
3.1 差异化 ..... 21

3.1.1 Notation ..... 22
3.1.1 符号 ..... 22

3.1.2 Standard Results ..... 23
3.1.2 标准结果 ..... 23

3.1.3 Product rule ..... 23
3.1.3 产品规则 ..... 23

3.1.4 Chain rule ..... 23
3.1.4 链式法则 ..... 23

3.1.5 Quotient rule ..... 24
3.1.5 商规则 ..... 24

3.1.6 Stationary points in 1D ..... 24
3.1.6 1D 中的静止点 ..... 24

3.1.7 Partial derivatives ..... 25
3.1.7 偏导数 ..... 25

3.1.8 Stationary points in 2 dimensions ..... 25
3.1.8 二维空间中的静止点 ..... 25

3.1.9 Taylor Series ..... 26
3.1.9 泰勒级数 ..... 26

3.2 Integration ..... 27
3.2 集成 ..... 27

3.2.1 Finding Integrals ..... 29
3.2.1 寻找积分 ..... 29

4 Complex Numbers ..... 32
4 复数 ..... 32

4.1 Motivation ..... 32
4.1 动机 ..... 32

4.1.1 Graphical concept ..... 32
4.1.1 图形概念 ..... 32

4.2 Definition ..... 33
4.2 定义 ..... 33

4.3 Complex Plane ..... 34
4.3 复平面 ..... 34

4.4 Addition/Subtraction ..... 34
4.4 加法/减法 ..... 34

4.5 Multiplication ..... 35
4.5 乘法 ..... 35

4.6 Conjugates ..... 35
4.6 结合物 ..... 35

4.7 Division ..... 37
4.7 分割 ..... 37

4.8 Polar Form ..... 38
4.8 极坐标形式 ..... 38

4.9 Exponential Notation ..... 39
4.9 指数表示法 ..... 39

4.10 Application to waves ..... 40
4.10 应用于波浪 ..... 40

4.10.1 Amplitude and phase ..... 41
4.10.1 幅度和相位 ..... 41

4.10.2 Complex solution to the wave equation ..... 43
4.10.2 波动方程的复杂解..... 43

5 Error analysis ..... 45
5 错误分析 ..... 45

5.1 Plus/Minus Notation ..... 46
5.1 正负符号表示法 ..... 46

5.2 Propagation of errors ..... 46
5.2 误差的传播.....46

5.3 Comparison with "worst case" scenario? ..... 47
5.3 与“最坏情况”进行比较?..... 47

5.4 Normal Distribution ..... 47
5.4 正态分布 ..... 47

5.5 Central limit theorem ..... 48
5.5 中心极限定理 ..... 48

5.6 Confidence Intervals ..... 49
5.6 置信区间 ..... 49

1 Basic Skills 1 基本技能

This document contains notes on basic mathematics. There are links to the corresponding Leeds University Library skills@Leeds page, in which there are subject notes, videos and examples.
本文档包含有关基本数学的笔记。其中包含指向对应的利兹大学图书馆技能@利兹页面的链接,页面中包含主题笔记、视频和示例。
If you require more in-depth explanations of these concepts, you can visit the Wolfram Mathworld website:
如果您需要对这些概念进行更深入的解释,您可以访问沃尔夫勒姆数学世界网站:
Wolfram link (http://mathworld.wolfram.com/)
Wolfram 链接 (http://mathworld.wolfram.com/)
  • Algebra (Expanding brackets, Factorising) :
    代数(展开括号,因式分解):
Library link  图书馆链接
(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/algebra/).
http://library.leeds.ac.uk/tutorials/maths-solutions/pages/algebra/。

- Fractions : - 分数:

(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/numeracy/fractions.html).
http://library.leeds.ac.uk/tutorials/maths-solutions/pages/numeracy/fractions.html.
  • Indices and Powers :
    指数和幂:
(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/numeracy/indices.html).
http://library.leeds.ac.uk/tutorials/maths-solutions/pages/numeracy/indices.html.

- Vectors : - 向量:

(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/mechanics/vectors.html).
http://library.leeds.ac.uk/tutorials/maths-solutions/pages/mechanics/vectors.html.
  • Trigonometry and geometry :
    三角学和几何学:
(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/trig_geom/).
http://library.leeds.ac.uk/tutorials/maths-solutions/pages/trig_geom/。
  • Differentiation and Integration :
    微积分:差异化和整合
Library link  图书馆链接
(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/calculus/).
http://library.leeds.ac.uk/tutorials/maths-solutions/pages/calculus/。

1.1 Practice Questions 1.1 练习题

There are practice equations available online to accompany these notes.
这些笔记配有在线练习方程。

2 Linear Algebra 2 线性代数

Wolfram link (http://mathworld.wolfram.com/LinearAlgebra.html)
Wolfram 链接(http://mathworld.wolfram.com/LinearAlgebra.html)

2.1 Matrices and Vectors
2.1 矩阵和向量

Library link (http://library.leeds.ac.uk/tutorials/maths-solutions/pages/mechanics/vectors.html)
图书馆链接 (http://library.leeds.ac.uk/tutorials/maths-solutions/pages/mechanics/vectors.html)

2.1.1 Definitions 2.1.1 定义

A matrix is a rectangular array of numbers enclosed in brackets. These numbers are called
矩阵是一组用括号括起来的数字的矩形数组。这些数字被称为
e.g. 例如。
Matrix has 2 rows and 3 columns.
矩阵 有 2 行 3 列。
A row vector is a matrix with a single row:
一行向量是具有单行的矩阵:
e.g. 例如。
Whereas a column vector is a matrix with a single column:
一个列向量是一个只有一列的矩阵:
e.g. 例如。
The size of a matrix is defined by where is the number of rows and is the number of columns. Matrix , as defined in equation 1 , is a matrix.
矩阵的大小由 定义,其中 是行数, 是列数。矩阵 ,如方程 1 中定义的,是一个 矩阵。
An element of a matrix can be described by its row position and column position. For ex-
矩阵的一个元素可以通过其行位置和列位置来描述。例如-

ample: the top left element in matrix , equal to 1 , is in row 1 and column 1 and can be labelled as element ; the element in the column of row 1 , equal to 3 , is labelled as . A general element is located in row and column (see equation 4 for a further example).
在矩阵 中,左上角的元素等于 1,位于第 1 行第 1 列,可以标记为元素 ;第 1 行的 列元素等于 3,标记为 。一般元素 位于第 行和第 列(参见方程 4 以获取更多示例)。

2.1.2 Notation 2.1.2 符号

There are different types of notation for matrices and vectors that you may encounter in text books. Below are some examples:
矩阵和向量的表示法有不同类型,你可能在教科书中遇到。以下是一些示例:
Matrix 矩阵
italics 斜体
bold, italics 粗体,斜体
double underline, italics
双下划线,斜体
Vector 矢量
italics 斜体
top arrow, italics 顶部箭头,斜体
single underline, italics
单下划线,斜体
bold 粗体

2.1.3 Addition 2.1.3 加法

Wolfram link (http://mathworld.wolfram.com/MatrixAddition.html)
Wolfram 链接(http://mathworld.wolfram.com/MatrixAddition.html)
Video link (http://www.youtube.com/watch?v=FX4C-JpTFgY)
视频链接 (http://www.youtube.com/watch?v=FX4C-JpTFgY)
Two matrices (or vectors) of the same size may be added together, element by element. For instance, if we have two matrices and :
两个相同大小的矩阵(或向量) 可以逐个元素相加。例如,如果我们有两个矩阵
then, 那么,

2.1.4 Subtraction 2.1.4 减法

Similar to addition, corresponding elements in and are subtracted from each other:
与加法类似, 中对应的元素相互相减:

2.1.5 Multiplication by a scalar
2.1.5 标量乘法

If is a number (i.e. a scalar) and is a matrix, then is also a matrix with entries
如果 是一个数字(即标量),而 是一个矩阵,则 也是一个具有条目的矩阵

2.1.6 Multiplication of two matrices
两个矩阵的乘法

Wolfram link (http://mathworld.wolfram.com/MatrixMultiplication.html)
Wolfram 链接(http://mathworld.wolfram.com/MatrixMultiplication.html)
This is non-trivial and is governed by a special rule. Two matrices , where is of size , and of size , can only be multiplied if , i.e. the number of columns in must match the number of rows in . The matrix produced has size , with each entry being the dot (or scalar) product (see section 2.1.10) of a whole row in by a whole column in .
这是一个非平凡的问题,受特殊规则约束。两个矩阵 ,其中 的大小为 的大小为 ,只有在 的情况下才能相乘,即 中的列数必须与 中的行数相匹配。生成的矩阵大小为 ,每个条目都是 中整行与 中整列的点(或标量)积(参见第 2.1.10 节)。

e.g. if 例如
then 然后
Formally, if 正式地,如果

Aside 一旁

When using Matlab (or octave), two matrices can be multiplied in an element-wise sense. This is NOT the same as described above.
在使用 Matlab(或 Octave)时,可以以逐元素方式相乘两个矩阵。这与上面描述的不同。

2.1.7 Motivation for matrix-matrix multiplication
2.1.7 矩阵矩阵乘法的动机

To understand why we may need to perform matrix-matrix multiplication, consider two customers of a repair garage, Peter and Alex, who require a number of car parts for each of their vehicles. Peter requires litre engine and 2 doors, whereas Alex requires litre engine and 4 doors. All the parts require a certain number of screws and bolts. But how many total screws and bolts do Peter and Alex need?
为了理解为什么我们可能需要执行矩阵-矩阵乘法,请考虑修车厂的两位顾客 Peter 和 Alex,他们各自的车辆需要一些汽车零件。Peter 需要 升引擎和 2 扇门,而 Alex 需要 升引擎和 4 扇门。所有零件都需要一定数量的螺丝和螺栓。但 Peter 和 Alex 总共需要多少螺丝和螺栓?
We can present the quantity of each car part that Peter and Alex need in a table:
我们可以用表格来展示彼得和亚历克斯需要的每个汽车零件的数量:
3 litre engine 3 升引擎 5 litre engine 5 升引擎 Doors 
Peter 彼得 1 0 2
Alex 亚历克斯 0 1 4
or as the matrix, :
或者作为矩阵,
The number of screws and bolts for each car part are expressed in the table:
每个汽车零件的螺丝和螺栓数量在表中表示:
bolts 螺栓 screws 螺丝钉
3 litre 3 升 3 4
5 litre 5 升 1 8
doors  2 1
or can be expressed as matrix, :
或者可以表示为矩阵,
Using simple addition we can find out how many screws and bolts are needed.
使用简单的加法,我们可以找出需要多少螺丝和螺栓。
  1. How many bolts are needed for Peter's car parts?
    彼得的汽车零件需要多少个螺栓?
.
  1. How many bolts are needed for Alex's car parts?
    Alex 的汽车零件需要多少个螺栓?
  1. How many screws are needed for Peter's car parts?
    彼得的汽车零件需要多少颗螺丝?
Or we can use matrix multiplication to get all four scenarios:
或者我们可以使用矩阵乘法得到所有四种情况:

2.1.8 Matrix-vector multiplication
2.1.8 矩阵-向量乘法

Since a vector is a special case of a matrix, this is simply a special case of the matrix-matrix multiplication we have already discussed. Consider multiplying a column vector of length by a matrix of size ,
由于向量是矩阵的特殊情况,这只是我们已经讨论过的矩阵乘法的特殊情况。考虑将长度为 的列向量乘以大小为 的矩阵,
e.g. 例如。
which results in a column vector of length and in this case .
这导致一个长度为 的列向量,在这种情况下

2.1.9 Special Matrices 2.1.9 特殊矩阵

Identity Matrix, The identity matrix, , of size , is defined in equation 12 .
单位矩阵, 大小为 的单位矩阵, ,在方程式 12 中定义。
i.e. if  即如果
This is a special case of a diagonal matrix possessing non-zero entries only on its diagonal e.g.
这是对角矩阵的一个特殊情况,只在对角线上具有非零条目,例如。
If is a square matrix, then the identity matrix has the special property that:
如果 是一个方 矩阵,则单位矩阵 具有特殊性质,即:
NB: is the equivalent of 1 in scalar arithmetic i.e. .
NB: 是标量算术中的 1 的等价物,即
Transpose, : If is a matrix then the transpose of , denoted , is a matrix found by swapping rows and columns of ,
转置, :如果 是一个 矩阵,那么 的转置,记为 ,是通过交换 的行和列找到的 矩阵
e.g. 例如。
Inverse matrix, If is an matrix, sometimes (see later) there exists another matrix called the inverse of , written , such that
逆矩阵,如果 是一个 矩阵,有时(见后文)存在另一个矩阵称为 的逆矩阵,记作 ,使得
NB: For scalar numbers, is the inverse of when considering multiplication, since
NB:对于标量数字,在考虑乘法时, 的倒数
Clearly when this breaks down and has no inverse - this is also true when dealing with some matrices.
明显地崩溃并且 没有逆时 - 处理某些矩阵时也是如此。

2.1.10 Scalar products and orthogonality
2.1.10 标量积和正交

The scalar product (or dot product, or inner product) of two column vectors of length , where and , is
两个长度为 的列向量(或点积,或内积)的数量积,其中 ,是
This can also be written as ; that is, the product of a row vector of length with a column vector of length . Two vectors are said to be orthogonal if their scalar product is zero.
这也可以写成 ;也就是说,长度为 的行向量与长度为 的列向量的乘积。如果两个向量的数量积为零,则称它们是正交的。

2.2 Linear Systems 2.2 线性系统

Wolfram link (http://mathworld.wolfram.com/LinearSystemofEquations.html)
Wolfram 链接(http://mathworld.wolfram.com/LinearSystemofEquations.html)
Video link (http://www.youtube.com/watch? )
视频链接(http://www.youtube.com/watch?
A linear system of equations such as
线性方程组如
can be written as
可以写成
as can be verified by multiplying out the left hand side. When solving the linear system , (where is a matrix, is the vector and is a vector of numbers) two cases can arise:
左侧展开后可以验证。解线性系统 时(其中 是矩阵, 是向量 是数字向量),可能出现两种情况:

i) exists. i) 存在。
ii) doesn't exist. There is then no solution in general.
ii) 不存在。通常情况下没有解决方案。

2.3 Determinants 2.3 行列式

Wolfram link (http://mathworld.wolfram.com/Determinant.html)
Wolfram 链接(http://mathworld.wolfram.com/Determinant.html)
Video link (http://www.youtube.com/watch?v=23LLB9mNJvc)
视频链接(http://www.youtube.com/watch?v=23LLB9mNJvc)
How do we know when exists? One method is to calculate the determinant of , written det or . The determinant is a single number that contains enough information about to determine whether it is invertible.
我们如何知道 存在?一种方法是计算 的行列式,写作 det 。行列式是一个包含足够关于 的信息的单个数字,可以确定它是否可逆。
determinants: In the case, if
决定因素:在 情况下,如果
then . For example, for the linear system given by
然后 。例如,对于给定的线性系统
the determinant of the coefficient matrix is
系数矩阵的行列式是
The matrix is therefore invertible (see section 2.3.1) and so the solution exists.
矩阵 是可逆的(见第 2.3.1 节),因此解存在。
As another example, consider
作为另一个例子,请考虑
The determinant of the coefficient matrix is
系数矩阵的行列式是
Therefore, has no inverse and so no solution exists. This can also be seen in the fact that it is not possible that