# 线性代数作业代写linear algebra代考|THREE–DIMENSIONAL GEOMETRY

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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|Introduction

In this chapter we present a vector-algebra approach to three-dimensional geometry. The aim is to present standard properties of lines and planes, with minimum use of complicated three-dimensional diagrams such as those involving similar triangles. We summarize the chapter:

Points are defined as ordered triples of real numbers and the distance between points $P_{1}=\left(x_{1}, y_{1}, z_{1}\right)$ and $P_{2}=\left(x_{2}, y_{2}, z_{2}\right)$ is defined by the formula
$$P_{1} P_{2}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}} .$$
Directed line segments $\overrightarrow{A B}$ are introduced as three-dimensional column vectors: If $A=\left(x_{1}, y_{1}, z_{1}\right)$ and $B=\left(x_{2}, y_{2}, z_{2}\right)$, then
$$\overrightarrow{A B}=\left[\begin{array}{c} x_{2}-x_{1} \ y_{2}-y_{1} \ z_{2}-z_{1} \end{array}\right]$$
If $P$ is a point, we let $\mathbf{P}=\overrightarrow{O P}$ and call $\mathbf{P}$ the position vector of $P$.
With suitable definitions of lines, parallel lines, there are important geometrical interpretations of equality, addition and scalar multiplication of vectors.
(i) Equality of vectors: Suppose $A, B, C, D$ are distinct points such that no three are collinear. Then $\overrightarrow{A B}=\overrightarrow{C D}$ if and only if $\overrightarrow{A B} | \overrightarrow{C D}$ and $\overrightarrow{A C} | \overrightarrow{B D}$

Figure 8.2: Scalar multiplication of vectors.
The dot product $X \cdot Y$ of vectors $X=\left[\begin{array}{l}a_{1} \ b_{1} \ c_{1}\end{array}\right]$ and $Y=\left[\begin{array}{c}a_{2} \ b_{2} \ c_{2}\end{array}\right]$, is defined by
$$X \cdot Y=a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}$$
The length $|X|$ of a vector $X$ is defined by
$$|X|=(X \cdot X)^{1 / 2}$$
and the Cauchy-Schwarz inequality holds:
$$|X \cdot Y| \leq|X| \cdot|Y|$$
The triangle inequality for vector length now follows as a simple deduction:
$$|X+Y| \leq|X|+|Y|$$
Using the equation
$$A B=|\overrightarrow{A B}|$$
we deduce the corresponding familiar triangle inequality for distance:
$$A B \leq A C+C B$$

(i) 向量相等：假设一种,乙,C,D是不同的点，因此没有三个点是共线的。然后一种乙→=CD→当且仅当一种乙→|CD→和一种C→|乙D→

X⋅和=一种1一种2+b1b2+C1C2

|X|=(X⋅X)1/2
Cauchy-Schwarz 不等式成立：
|X⋅和|≤|X|⋅|和|

|X+和|≤|X|+|和|

# 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