# 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
• 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|Frames in the Plane

Before leaving the subject of motions in the plane we want to discuss another approach to defining them – one that will be especially powerful in higher dimensions.

Definition. A frame in $\mathbf{R}^{2}$ is a tuple $\mathrm{F}=\left(\mathbf{u}{1}, \mathbf{u}{2}, \mathbf{p}\right)$, where $\mathbf{p}$ is a point and $\mathbf{u}{1}$ and $\mathbf{u}{2}$ define an orthonormal basis of $\mathbf{R}^{2}$. If the ordered basis $\left(\mathbf{u}{1}, \mathbf{u}{2}\right)$ induces the standard orientation, then we shall call the frame an oriented frame. The lines determined by $\mathbf{p}$ and the direction vectors $\mathbf{u}{1}$ and $\mathbf{u}{2}$ are called the $x$-, respectively, $y$-axis of the frame $\mathrm{F}$. The point $\mathbf{p}$ is called the origin of the frame $\mathrm{F}$. $\left(\mathbf{e}{1}, \mathbf{e}{2}, \mathbf{0}\right)$ is called the standard frame of $\mathbf{R}^{2}$ To simplify the notation, we sometimes use $\left(\mathbf{u}{1}, \mathbf{u}{2}\right)$ to denote the frame $\left(\mathbf{u}{1}, \mathbf{u}{2}, \mathbf{0}\right)$. Frames can be thought of as defining a new coordinate system. See Figure 2.13. They can also be associated to a transformation in a natural way. If $\mathrm{F}=\left(\mathbf{u}{1}, \mathbf{u}{2}, \mathbf{p}\right)$ is a frame and if $\mathbf{u}{\mathrm{i}}=\left(\mathrm{u}{\mathrm{i} 1}, \mathrm{u}{\mathrm{i} 2}\right)$ and $\mathbf{p}=(\mathrm{m}, \mathrm{n})$, then define a map $\mathrm{T}{\mathrm{F}}$ by the equations \begin{aligned} &\mathrm{x}^{\prime}=\mathrm{u}{11} \mathrm{x}+\mathrm{u}{21} \mathrm{y}+\mathrm{m} \ &\mathrm{y}^{\prime}=\mathrm{u}{12} \mathrm{x}+\mathrm{u}{22} \mathrm{y}+\mathrm{n} . \end{aligned}

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

Definition. A map $\mathrm{S}: \mathbf{R}^{\mathrm{n}} \rightarrow \mathbf{R}^{\mathrm{n}}$ is called a similarity transformation, or simply a similarity, if $$|S(\mathbf{p}) S(\mathbf{q})|=r|\mathbf{p q}|$$ for all $\mathbf{p}, \mathbf{q} \in \mathbf{R}^{\mathrm{n}}$ and some fixed positive constant $r$. Clearly, motions are similarities, because they correspond to the case where $r$ is 1 in the definition. On the other hand, the map $\mathrm{S}(\mathbf{p})=2 \mathbf{p}$ is a similarity but not a motion. In fact, $\mathrm{S}$ an example of a simple but important class of similarities.

Definition. A map $\mathrm{R}: \mathbf{R}^{\mathrm{n}} \rightarrow \mathbf{R}^{\mathrm{n}}$ of the form $\mathrm{R}(\mathbf{p})=r \mathbf{p}, \mathrm{r}>0$, is called a radial transformation. 2.3.1. Theorem. Radial transformations are similarities. Proof. Exercise. The next theorem shows that similarities are not much more complicated than motions. 2.3.2. Theorem. If $S$ is a similarity, then $S=M R$, where $M$ is a motion and $R$ is a radial transformation. Conversely, any map of the form MR, where $M$ is a motion and $\mathrm{R}$ is a radial transformation, is a similarity.

Proof. This is easy because if we use the notation in the definitions for a similarity and a radial transformation, then $\mathrm{R}^{-1} \mathrm{~S}$ is a motion $\mathrm{M}$. 2.3.3. Corollary. Every similarity in the plane can be expressed by equations of the form \begin{aligned} &x^{\prime}=a x+b y+m \ &y^{\prime}=\pm(-b x+a y)+n \end{aligned} where $(a, b) \neq(0,0)$. (The $r$ in the definition of a similarity is $\sqrt{a^{2}+b^{2}}$ in this case.) Conversely, every map defined by such equations is a similarity.

## 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