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# 线性代数作业代写linear algebra代考|Introduction to linear equations

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|linear equations

A linear equation in $n$ unknowns $x_{1}, x_{2}, \cdots, x_{n}$ is an equation of the form
$$a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}=b,$$
where $a_{1}, a_{2}, \ldots, a_{n}, b$ are given real numbers.
For example, with $x$ and $y$ instead of $x_{1}$ and $x_{2}$, the linear equation $2 x+3 y=6$ describes the line passing through the points $(3,0)$ and $(0,2)$.
Similarly, with $x, y$ and $z$ instead of $x_{1}, x_{2}$ and $x_{3}$, the linear equation $2 x+3 y+4 z=12$ describes the plane passing through the points $(6,0,0),(0,4,0),(0,0,3)$.

A system of $m$ linear equations in $n$ unknowns $x_{1}, x_{2}, \cdots, x_{n}$ is a family of linear equations
\begin{aligned} a_{11} x_{1}+a_{12} x_{2}+\cdots+a_{1 n} x_{n} &=b_{1} \ a_{21} x_{1}+a_{22} x_{2}+\cdots+a_{2 n} x_{n} &=b_{2} \ & \vdots \ a_{m 1} x_{1}+a_{m 2} x_{2}+\cdots+a_{m n} x_{n} &=b_{m} . \end{aligned}
We wish to determine if such a system has a solution, that is to find out if there exist numbers $x_{1}, x_{2}, \cdots, x_{n}$ which satisfy each of the equations simultaneously. We say that the system is consistent if it has a solution. Otherwise the system is called inconsistent.

Note that the above system can be written concisely as
$$\sum_{j=1}^{n} a_{i j} x_{j}=b_{i}, \quad i=1,2, \cdots, m .$$
The matrix
$$\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \vdots & & & \vdots \ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right]$$
is called the coefficient matrix of the system, while the matrix
$$\left[\begin{array}{ccccc} a_{11} & a_{12} & \cdots & a_{1 n} & b_{1} \ a_{21} & a_{22} & \cdots & a_{2 n} & b_{2} \ \vdots & & & \vdots & \vdots \ a_{m 1} & a_{m 2} & \cdots & a_{m n} & b_{m} \end{array}\right]$$
is called the augmented matrix of the system.
Geometrically, solving a system of linear equations in two (or three) unknowns is equivalent to determining whether or not a family of lines (or planes) has a common point of intersection.

## 线性代数作业代写linear algebra代考|THE FIELD AXIOMS

1. $(a+b)+c=a+(b+c)$ for all $a, b, c$ in $F$;
2. $(a b) c=a(b c)$ for all $a, b, c$ in $F$;
3. $a+b=b+a$ for all $a, b$ in $F$;
4. $a b=b a$ for all $a, b$ in $F$;
5. there exists an element 0 in $F$ such that $0+a=a$ for all $a$ in $F$;
6. there exists an element 1 in $F$ such that $1 a=a$ for all $a$ in $F$;
1. to every $a$ in $F$, there corresponds an additive inverse $-a$ in $F$, satisfying
$$a+(-a)=0$$
2. to every non-zero $a$ in $F$, there corresponds a multiplicative inverse $a^{-1}$ in $F$, satisfying
$$a a^{-1}=1 ;$$
3. $a(b+c)=a b+a c$ for all $a, b, c$ in $F$;
4. $0 \neq 1$.
With standard definitions such as $a-b=a+(-b)$ and $\frac{a}{b}=a b^{-1}$ for $b \neq 0$, we have the following familiar rules:
\begin{aligned} -(a+b) &=(-a)+(-b), \quad(a b)^{-1}=a^{-1} b^{-1} \ -(-a) &=a, \quad\left(a^{-1}\right)^{-1}=a ; \ -(a-b) &=b-a, \quad\left(\frac{a}{b}\right)^{-1}=\frac{b}{a} \ \frac{a}{b}+\frac{c}{d} &=\frac{a d+b c}{b d} \ \frac{a}{b} \frac{c}{d} &=\frac{a c}{b d} ; \ \frac{a b}{a c} &=\frac{b}{c}, \frac{a}{\left(\frac{b}{c}\right)}=\frac{a c}{b} \ -(a b) &-(-a) b-a(-b) \ -\left(\frac{a}{b}\right) &=\frac{-a}{b}=\frac{a}{-b} \ 0 a &=0 ; \ (-a)^{-1} &=-\left(a^{-1}\right) . \end{aligned}
Fields which have only finitely many elements are of great interest in many parts of mathematics and its applications, for example to coding theory. It is easy to construct fields containing exactly $p$ elements, where $p$ is a prime number. First we must explain the idea of modular addition and modular multiplication. If $a$ is an integer, we define $a(\bmod p)$ to be the least remainder on dividing a by $p$ : That is, if $a=b p+r$, where $b$ and $r$ are integers and $0 \leq r<p$, then $a(\bmod p)=r$.

## 线性代数作业代写LINEAR ALGEBRA代考|LINEAR EQUATIONS

∑j=1n一种一世jXj=b一世,一世=1,2,⋯,米.

[一种11一种12⋯一种1n 一种21一种22⋯一种2n ⋮⋮ 一种米1一种米2⋯一种米n]

[一种11一种12⋯一种1nb1 一种21一种22⋯一种2nb2 ⋮⋮⋮ 一种米1一种米2⋯一种米nb米]

## 线性代数作业代写LINEAR ALGEBRA代考|THE FIELD AXIOMS

1. (一种+b)+C=一种+(b+C)对所有人一种,b,C在F;
2. (一种b)C=一种(bC)对所有人一种,b,C在F;
3. 一种+b=b+一种对所有人一种,b在F;
4. 一种b=b一种对所有人一种,b在F;
5. 中存在元素 0F这样0+一种=一种对所有人一种在F;
6. 中存在元素 1F这样1一种=一种对所有人一种在F;
1. 对每个一种在F, 对应一个加法逆−一种在F, 满足
一种+(−一种)=0
2. 对每个非零一种在F, 对应一个乘法逆一种−1在F, 满足
一种一种−1=1;
3. 一种(b+C)=一种b+一种C对所有人一种,b,C在F;
4. 0≠1.只有有限多个元素的领域在数学的许多部分及其应用中都非常有趣，例如编码理论。构造精确包含的字段很容易p元素，其中p是一个素数。首先我们必须解释模加和模乘的概念。如果一种是一个整数，我们定义一种(反对p)是除以 a 的最小余数p: 也就是说，如果一种=bp+r， 在哪里b和r是整数和0≤r<p， 然后一种(反对p)=r.

# 计量经济学代写

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions