# 线性代数网课代修|计算机图形学代写Computer Graphics代考|MATH240

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• 逼近论

## 线性代数作业代写linear algebra代考|Rigid Motions in the Plane

2.2.6.1. Lemma. Every rotation $\mathrm{R}$ of the plane can be expressed in the form $\mathrm{R}=$ $R_{0} T_{1}=T_{2} R_{0}$, where $R_{0}$ is a rotation about the origin and $T_{1}$ and $T_{2}$ are translations. Conversely, if $R_{0}$ is any rotation about the origin through a nonzero angle and if $T$ is a translation of the plane, then both $R_{0} T$ and $T_{0}$ are rotations.

Proof. Suppose that $\mathrm{R}=\mathrm{TR}{0} \mathrm{~T}^{-1}$, where $\mathrm{R}{0}$ is a rotation about the origin and $\mathrm{T}$ is a translation. By Theorem 2.2.4.2 we can move the translations to either side of $\mathrm{R}{0}$, which proves the first part of the lemma. The other part can be proved by showing that certain equations have unique solutions. For example, to show that $\mathrm{TR}{0}$ is a rotation, one assumes that it is a rotation about some point $(a, b)$ and tries to solve the equations
$$(x-a) \cos \theta-(y-b) \sin \theta+a=x \cos \theta-y \sin \theta+c$$ for a and b. The details are left as an exercise.
2.2.6.2. Theorem. The set of all translations and rotations of the plane is a subgroup of the group of all motions. The set of rotations by itself is not a group.

Proof. To prove the theorem one uses Lemma 2.2.6.1 to show that the composites of translations and rotations about an arbitrary point are again either a translation or a rotation.

Definition. A motion of the plane that is a composition of translations and/or rotations is called a rigid motion or displacement.

## 线性代数作业代写linear algebra代考|Summary for Motions in the Plane

We have defined motions and have shown that a motion of the plane is completely specified by what it does to three noncollinear points and that it can be described in terms of three very simple motions, namely, translations, rotations, and reflections. To understand such motions it suffices to have a good understanding of these three primitive types.

Planar motions are either orientation preserving or orientation reversing with rigid motions being the orientation-preserving ones. Reflections are orientation reversing. Another way to describe a planar motion is as a rigid motion or the composition of a rigid motion and a single reflection. In fact, we may assume that the reflection, if it is needed, is just the reflection about the $\mathrm{x}$-axis.

Combining various facts we know, it is now very casy to describe the equation of an arbitrary motion of the plane.
2.2.7.1. Theorem. Every motion $\mathrm{M}$ of the plane is defined by equations of the form
\begin{aligned} &x^{\prime}=a x+b y+c \ &y^{\prime}=\pm(-b x+a y)+d \end{aligned}
where $a^{2}+b^{2}=1$. Conversely, every such pair of equations defines a motion.
Proof. Let $\mathrm{M}(\mathbf{0})=(\mathrm{c}, \mathrm{d})$ and define a translation $\mathrm{T}$ by $\mathrm{T}(\mathbf{P})=\mathbf{P}+(\mathrm{c}, \mathrm{d})$. Let $\mathrm{M}^{\prime}=$ $\mathrm{T}^{-1} \mathrm{M}$. Then $\mathrm{M}=\mathrm{TM}^{\prime}$ and $\mathrm{M}^{\prime}$ fixes the origin.
Case 1. $\mathrm{M}$ is orientation preserving.
In this case $\mathbf{M}^{\prime}$ is orientation preserving and must be a rotation about the origin through some angle $\theta$ (Theorem 2.2.6.9). Let $\mathrm{a}=\cos \theta$ and $\mathrm{b}=-\sin \theta$. Clearly the equation for M has the desired form.

## 线性代数作业代写linear algebra代考|Rigid Motions in the Plane

2.2.6.1。引理。每转一圈 $R$ 平面的可表示为 $R=R_{0} T_{1}=T_{2} R_{0}$ ， 在哪里 $R_{0}$ 是关于原点的 旋转，并且 $T_{1}$ 和 $T_{2}$ 是翻译。相反，如果 $R_{0}$ 是通过非零角度围绕原点进行的任何旋转，如果 $T$ 是平面的平移，那么两者 $R_{0} T$ 和 $T_{0}$ 是旋转。

$$(x-a) \cos \theta-(y-b) \sin \theta+a=x \cos \theta-y \sin \theta+c$$

2.2.6.2。定理。平面的所有平移和旋转的集合是所有运动组的子组。这组旋转本身不是一 个组。

## 线性代数作业代写linear algebra代考|Summary for Motions in the Plane

2.2.7.1。定理。每一个动作M平面的由以下形式的方程定义
$$x^{\prime}=a x+b y+c \quad y^{\prime}=\pm(-b x+a y)+d$$

# 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