# 线性代数网课代修|计算机图形学代写Computer Graphics代考|COMP6370

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• 数值分析
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• 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|The Projective Plane

The informal discussion of linear equations and their solutions in the previous section led to homogeneous coordinates and suggested a new way of looking at points in the plane. We shall now develop these observations more rigorously. Although we are only interested in the projective line and plane for a while, we start off with some general definitions so that we do not have to repeat them for each dimension.
3.4.1. Lemma. The relation $\sim$ defined on the points $\mathbf{p}$ of $\mathbf{R}^{\mathrm{n}+1}-\mathbf{0}$ by $\mathbf{p} \sim \mathrm{cp}$, for $\mathrm{c} \neq$ 0 , is an equivalence relation.
Proof. This is an easy exercise.
Definition. The set of equivalence classes of $\mathbf{R}^{\mathrm{n}+1}-\mathbf{0}$ with respect to the relation $\sim$ defined in Lemma 3.4.1 is called the n-dimensional (real) projective space $\mathbf{P}^{\mathrm{n}}$. In more compact notation (see Section $5.4$ and the definition of a quotient space),
$$\mathbf{P}^{\mathrm{n}}=\left(\mathbf{R}^{\mathrm{n}+1}-\mathbf{0}\right) / \sim \text {. }$$
The special cases $\mathbf{P}^{1}$ and $\mathbf{P}^{2}$ are called the projective line and projective plane, respectively. If $\mathbf{P} \subset \mathbf{P}^{\mathrm{n}}$ and $\mathbf{P}=\left[\mathrm{x}{1}, \mathrm{x}{2}, \ldots, \mathrm{x}{\mathrm{n}+1}\right]$, then the numbers $\mathrm{x}{1}, \mathrm{x}{2}, \ldots, \mathrm{x}{\mathrm{n}+1}$ are called homogeneous coordinates of $\mathbf{P}$. One again typically uses the expression ” $\left(\mathrm{x}{1}, \mathrm{x}{2}, \ldots, \mathrm{x}_{\mathrm{n}+1}\right)$ are homogeneous coordinates for $\mathbf{P}^{\prime \prime}$ in that case.

Note that $\mathbf{P}^{0}$ consists of the single point [1]. We can think of points in $\mathbf{P}^{1}$ or $\mathbf{P}^{2}$ as equivalence classes of solutions to (3.13) or (3.15), or alternatively, as the set of lines through the origin in $\mathbf{R}^{2}$ or $\mathbf{R}^{3}$, respectively. Other characterizations of the abstract spaces $\mathbf{P}^{\mathrm{n}}$ will be given in Section 5.9. There are actually many ways to introduce coordinates for their points. In the next section we shall see how this can be done for $\mathbf{P}^{1}$ and $\mathbf{P}^{2}$.
It is easy to check that the maps $\mathbf{P}^{0} \rightarrow \mathbf{P}^{1} \quad, \quad \mathbf{P}^{1} \quad \rightarrow \quad \mathbf{P}^{2}$
$[\mathrm{X}] \rightarrow[0, \mathrm{X}] \quad[\mathrm{X}, \mathrm{Y}] \rightarrow[\mathrm{X}, 0, \mathrm{Y}]$
and
are one-to-one. Therefore, by identifying the corresponding points, we shall think of these maps as inclusion maps and get a commutative diagram
\begin{aligned} &\mathbf{P}^{0} \subset \mathbf{P}^{1} \subset \mathbf{P}^{2} \ &| \quad \cup \cup \ &\mathbf{R}^{0} \subset \mathbf{R}^{1} \subset \mathbf{R}^{2} \end{aligned}

## 线性代数作业代写linear algebra代考|Analytic Properties of the Projective Plane

This section describes some analytic properties of $\mathbf{P}^{2}$.
3.4.1.1. Theorem. Three distinct points $\left[\mathrm{X}{1}, \mathrm{Y}{1}, \mathrm{Z}{1}\right],\left[\mathrm{X}{2}, \mathrm{Y}{2}, \mathrm{Z}{2}\right]$, and $\left[\mathrm{X}{3}, \mathrm{Y}{3}, \mathrm{Z}{3}\right]$ of $\mathbf{P}^{2}$ are collinear if and only if $$\left|\begin{array}{lll} \mathrm{X}{1} & \mathrm{Y}{1} & \mathrm{Z}{1} \ \mathrm{X}{2} & \mathrm{Y}{2} & \mathrm{Z}{2} \ \mathrm{X}{3} & \mathrm{Y}{3} & \mathrm{Z}{3} \end{array}\right|=0$$
Proof. We basically have to find numbers a, b, and c, not all zero, so that
$$a X_{i}+b Y_{i}+c Z_{i}=0, \quad i=1,2,3 .$$
The theorem is now an easy consequence of basic facts about when such systems of equations admit nontrivial solutions.
3.4.1.2. Corollary. The line in $\mathbf{P}^{2}$ determined by two distinct points $\left[X_{1}, Y_{1}, Z_{1}\right]$ and $\left[\mathrm{X}{2}, \mathrm{Y}{2}, \mathrm{Z}{2}\right]$ has equation $$\left|\begin{array}{ll} \mathrm{Y}{1} & \mathrm{Z}{1} \ \mathrm{Y}{2} & \mathrm{Z}{2} \end{array}\right| \mathrm{X}+\left|\begin{array}{ll} \mathrm{Z}{1} & \mathrm{X}{1} \ \mathrm{Z}{2} & \mathrm{X}{2} \end{array}\right| \mathrm{Y}+\left|\begin{array}{cc} \mathrm{X}{1} & \mathrm{Y}{1} \ \mathrm{X}{2} & \mathrm{Y}{2} \end{array}\right| \mathrm{Z}=0 .$$ Proof. Simply apply Theorem 3.4.1.1 to points $[\mathrm{X}, \mathrm{Y}, \mathrm{Z}],\left[\mathrm{X}{1}, \mathrm{Y}{1}, \mathrm{Z}{1}\right]$, and $\left[\mathrm{X}{2}, \mathrm{Y}{2}, \mathrm{Z}{2}\right]$ and expand the determinant in the theorem by minors using the top row of the matrix. 3.4.1.3. Theorem. If the lines $\mathbf{L}{1}$ in $\mathbf{P}^{2}$ defincd by cquations
$$a_{\mathrm{i}} \mathrm{X}+\mathrm{b}{\mathrm{i}} \mathrm{Y}+\mathrm{c}{\mathrm{i}} \mathrm{Z}=0, \quad \mathrm{i}=1,2,$$
are distinct, then they intersect in the point $\left[\left(\mathrm{a}{1}, \mathrm{~b}{1}, \mathrm{c}{1}\right) \times\left(\mathrm{a}{2}, \mathrm{~b}{2}, \mathrm{c}{2}\right)\right]$.
Proof. This follows from the fact that the cross product of two vectors is orthogonal to both of the vectors.
3.4.1.4. Theorem. Let $\mathbf{L}$ be the line in $\mathbf{P}^{\text {? }}$ determined by two distinct points $\mathbf{P}{1}=\left[\mathbf{p}{1}\right]$ and $\mathbf{P}{2}=\left[\mathbf{p}{2}\right], \mathbf{p}_{\mathrm{i}} \in \mathbf{R}^{3}$.

## 线性代数作业代写linear algebra代考|The Projective Plane

3.4.1。引理。关系 在点上定义 $\mathbf{p}$ 的 $\mathbf{R}^{\mathbf{n}+1}-\mathbf{0}$ 经过 $\mathbf{p} \sim \mathrm{cp}$ ，为了 $\mathbf{c} \neq 0$ ，是等价关系。 证明。这是一个简单的练习。

$$\mathbf{P}^{\mathrm{n}}=\left(\mathbf{R}^{\mathrm{n}+1}-\mathbf{0}\right) / \sim .$$

$[\mathrm{X}] \rightarrow[0, \mathrm{X}] \quad[\mathrm{X}, \mathrm{Y}] \rightarrow[\mathrm{X}, 0, \mathrm{Y}]$

$$\mathbf{P}^{0} \subset \mathbf{P}^{1} \subset \mathbf{P}^{2} \quad \mid \quad \cup \cup \mathbf{R}^{0} \subset \mathbf{R}^{1} \subset \mathbf{R}^{2}$$

## 线性代数作业代写linear algebra代考|Analytic Properties of the Projective Plane

3.4.1.1。定理。三个不同的点 $[\mathrm{X} 1, \mathrm{Y} 1, \mathrm{Z} 1],[\mathrm{X} 2, \mathrm{Y} 2, \mathrm{Z} 2]$ ，和 $[\mathrm{X} 3, \mathrm{Y} 3, \mathrm{Z} 3]$ 的 $\mathbf{P}^{2}$ 共线当且 仅当
$\left|\begin{array}{lllllll}\mathrm{X} 1 & \mathrm{Y} 1 & \mathrm{Z} 1 \mathrm{X} 2 & \mathrm{Y} 2 & \mathrm{Z} 2 \mathrm{X} 3 & \mathrm{Y} 3 & \mathrm{Z} 3\end{array}\right|=0$

$$a X_{i}+b Y_{i}+c Z_{i}=0, \quad i=1,2,3$$

3.4.1.2。推论。线在 $\mathbf{P}^{2}$ 由两个不同的点决定 $\left[X_{1}, Y_{1}, Z_{1}\right]$ 和 $[\mathrm{X} 2, \mathrm{Y} 2, \mathrm{Z} 2]$ 有方程
$|\mathrm{Y} 1 \quad \mathrm{Z} 1 \mathrm{Y} 2 \quad \mathrm{Z} 2| \mathrm{X}+|\mathrm{Z} 1 \quad \mathrm{X} 1 \mathrm{Z} 2 \quad \mathrm{X} 2| \mathrm{Y}+|\mathrm{X} 1 \quad \mathrm{Y} 1 \mathrm{X} 2 \quad \mathrm{Y} 2| \mathrm{Z}=0$

$$a_{\mathrm{i}} \mathrm{X}+\mathrm{biY}+\mathrm{ciZ}=0, \quad \mathrm{i}=1,2,$$

3.4.1.4。定理。让 $\mathbf{L}$ 排队 $\mathbf{P}^{?}$ 由两个不同的点决定 $\mathbf{P} 1=[\mathbf{p} 1]$ 和 $\mathbf{P} 2=[\mathbf{p} 2], \mathbf{p}_{\mathrm{i}} \in \mathbf{R}^{3}$.

# 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