# 线性代数网课代修|计算机图形学代写Computer Graphics代考|COMP7204

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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|Affine Transformations

Definition. A one-to-one and onto mapping $\mathrm{T}: \mathbf{R}^{\mathrm{n}} \rightarrow \mathbf{R}^{\mathrm{n}}$ that maps lines onto lines is called an affine transformation.

Actually, one can characterize affine transformations in a slightly stronger fashion.
2.4.1. Theorem. Any one-to-one and onto map of $\mathbf{R}^{\mathrm{n}}$ onto itself that preserves collinearity is an affine transformation.

Proof. The only thing that needs to be shown is that lines get mapped onto lines. This is shown in a way similar to what was done in the proof of Lemma 2.2.4 and left as an exercise.
2.4.2. Theorem. The set of affine transformations in $\mathbf{R}^{\mathrm{n}}$ forms a group that contains the similarities as a subgroup.
Proof. Exercise.
Affine transformations, like motions and similarities, have a simple analytic description. Before we get to the main result for these maps in the plane, we analyze transformations with equations of the form \begin{aligned} &x^{\prime}=a x+b y+m \ &y^{\prime}=c x+d y+n \end{aligned}
where
$$\left|\begin{array}{ll} a & b \ c & d \end{array}\right| \neq 0 .$$

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

Definition. Let $\mathbf{v}$ be a nonzero vector in $\mathbf{R}^{\mathrm{n}}$ and let $\Omega$ be the family of parallel lines with direction vector $\mathbf{v}$. Let $\mathbf{L}_{\mathbf{p}}$ denote the line in $\Omega$ through the point $\mathbf{p}$. If $\mathbf{X}$ is a hyperplane in $\mathbf{R}^{\mathrm{n}}$ not parallel to $\mathbf{v}$, then define a map $$\pi_{\Omega}: \mathbf{R}^{\mathrm{n}} \rightarrow \mathbf{X}$$
by
$$\pi_{\Omega}(\mathbf{p})=\mathbf{L}{\mathbf{p}} \cap \mathbf{X} .$$ The map $\pi{\Omega}$ is called the parallel projection of $\boldsymbol{R}^{n}$ onto the plane $\boldsymbol{X}$ parallel to $\boldsymbol{v}$. If $\mathbf{v}$ is orthogonal to $\mathbf{X}$, then $\pi_{\Omega}$ is called the orthogonal or orthographic projection of $\boldsymbol{R}^{n}$ onto the plane $\boldsymbol{X}$; otherwise, it is called an oblique parallel projection. In general, if $\mathbf{X}$ and $\mathbf{Y}$ are any subsets of $\mathbf{R}^{\mathrm{n}}$, then the map that sends $\mathbf{p}$ in $\mathbf{X}$ to $\mathbf{L}_{\mathbf{p}} \cap \mathbf{Y}$ in $\mathbf{Y}$ (wherever it is defined) is called the parallel projection of $\boldsymbol{X}$ to $\boldsymbol{Y}$.

Figure $2.22$ shows a parallel projection of a line $\mathbf{L}$ onto a line $\mathbf{L}^{\prime}$ and Figure 2.23, a parallel projection of a plane $\mathbf{X}$ onto a plane $\mathbf{X}^{\prime}$. Note that the ratio of distances is preserved in the case of parallel projections of a line onto another line. What this means is that, referring to Figure $2.22$, the ratio
$$\frac{|\mathbf{A B}|}{\left|\mathbf{A}^{\prime} \mathbf{B}^{\prime}\right|}$$
is independent of $\mathbf{A}$ and $\mathbf{B}$. This is not the case for parallel projections of one plane onto another. For example, in Figure $2.23$ the ratios $$\frac{|\mathbf{A B}|}{\left|\mathbf{A}^{\prime} \mathbf{B}^{\prime}\right|} \text { and } \frac{|\mathbf{B C}|}{\left|\mathbf{B}^{\prime} \mathbf{C}^{\prime}\right|}$$
are probably not the same.

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

2.4.1。定理。任意一对一并上图 $\mathbf{R}^{\mathrm{n}}$ 保持共线性的自身是仿射变换。

2.4.2. 定理。仿射变换的集合 $\mathbf{R}^{\mathrm{n}}$ 形成一个包含相似性的组作为子组。

$$x^{\prime}=a x+b y+m \quad y^{\prime}=c x+d y+n$$

$$\mid \begin{array}{lll} a & b c & d \mid \neq 0 \end{array}$$

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

$$\pi_{\Omega}(\mathbf{p})=\mathbf{L} \mathbf{p} \cap \mathbf{X} .$$

$$\frac{|\mathbf{A B}|}{\left|\mathbf{A}^{\prime} \mathbf{B}^{\prime}\right|}$$

$$\frac{|\mathbf{A B}|}{\left|\mathbf{A}^{\prime} \mathbf{B}^{\prime}\right|} \text { and } \frac{|\mathbf{B C}|}{\left|\mathbf{B}^{\prime} \mathbf{C}^{\prime}\right|}$$

# 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