# 线性代数网课代修|计算机图形学代写Computer Graphics代考|COMP3170

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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Beyond the Plane

Up to now, although some things applied to $\mathbf{R}^{\mathrm{n}}$, most of the details were specifically about transformations in the plane. The fact is that much of what we did generalizes to higher dimensions. We start with motions of $\mathbf{R}^{\mathrm{n}}$. 2.5.1. Theorem. Every motion $\mathbf{M}: \mathbf{R}^{\mathrm{n}} \rightarrow \mathbf{R}^{\mathrm{n}}$ can be expressed by equations of the form \begin{aligned} &x_{1}^{\prime}=a_{11} x_{1}+a_{12} x_{2}+\ldots+a_{1 n} x_{n}+c_{1} \ &x_{2}^{\prime}=a_{21} x_{1}+a_{22} x_{2}+\ldots+a_{2 n} x_{n}+c_{2} \end{aligned} $$\mathrm{x}{\mathrm{n}}^{\prime}=\mathrm{a}{\mathrm{n} 1} \mathrm{x}{1}+\mathrm{a}{\mathrm{n} 2} \mathrm{x}{2}+\ldots+\mathrm{a}{\mathrm{nn}} \mathrm{x}{\mathrm{n}}+\mathrm{c}{\mathrm{n}}$$ where $A_{M}=\left(a_{i j}\right)$ is an orthogonal matrix. Conversely, every such system of equations defines a motion.

Proof. The discussion in Section $2.2 .8$ on frames showed that the theorem is valid for motions in the plane. For the general case, assume without loss of generality that $\mathbf{M}(\mathbf{0})=\mathbf{0}$. The key facts are Theorem $2.2 .4 .1$, which says that $\mathrm{M}$ is a linear transformation (and hence is defined by a matrix), and Lemma 2.2.4.3, which says that $\mathrm{M}(\mathbf{u}) \cdot \mathrm{M}(\mathbf{v})=\mathbf{u} \cdot \mathbf{v}$, for all vectors $\mathbf{u}$ and $\mathbf{v}$. The rest of the proof simply involves analyzing the conditions $\mathrm{M}\left(\mathbf{e}{\mathrm{i}}\right) \cdot \mathrm{M}\left(\mathbf{e}{\mathrm{j}}\right)=\mathbf{e}{\mathrm{i}} \bullet \mathbf{e}{\mathrm{j}}=\delta_{\mathrm{ij}}$ and is left as an exercise (Exercise 2.5.1). In studying motions in the plane we made use of some important special motions, such as translations, rotations, and reflections. Translations already have a general definition. The natural generalization of the definition of a reflection is to replace lines by hyperplanes.

Definition. Let $\mathbf{X}$ be a hyperplane in $\mathbf{R}^{\mathrm{n}}$. Define a map $\mathrm{S}: \mathbf{R}^{\mathrm{n}} \rightarrow \mathbf{R}^{\mathrm{n}}$, called the reflection about the hyperplane $\boldsymbol{X}$, as follows: Let $\mathbf{A}$ be a point in $\mathbf{X}$ and let $\mathbf{N}$ be a normal vector for $\mathbf{X}$. If $\mathbf{P}$ is any point in $\mathbf{R}^{\mathrm{n}}$, then $\mathbf{S}(\mathbf{P})=\mathbf{P}+2 \mathbf{P Q}$, where $\mathbf{P Q}$ is the orthogonal projection of $\mathbf{P A}$ on $\mathbf{N}$. See Figure 2.25.

## 线性代数作业代写linear algebra代考|Motions in 3-Space

In this section we look at the mechanics of transforming objects in 3-space. This may not seem as easy as it was in the plane, but if we break the general problem into a sequence of simple primitive ones, then it will become easy again.

Rigid motions are composites of rotations and/or translations. Is is useful to have some alternate characterizations of rotations. The first characterization comes from the Principal Axis Theorem (Theorem 2.5.5), which says that every rotation is a rotation about an axis. Before we can make use of this way of looking at a rotation we must resolve an ambiguity that we alluded to in a comment immediately following Theorem 2.5.5. Suppose that $\mathbf{v}$ is a direction vector for the axis. If we consider a plane orthogonal to the axis of the rotation, the notion of counterclockwise for this plane, which is what is normally used to define the positive direction for an angle, will depend on whether we are looking down on this plane from a point on the axis in the $\mathbf{v}$ or $-\mathbf{v}$ direction. The only way that this ambiguity in the expression “a rotation about a line through a given angle” can be avoided is by requiring the line to be oriented. The axis-angle representation of a rotation: Here we represent a rotation by a triple $(\mathbf{p}, \mathbf{u}, \theta)$, where the point $\mathbf{p}$ and unit (direction) vector $\mathbf{u}$ specify the axis and $\theta$ is the angle of rotation determined according to the following rule:

The rotation orientation rule: Think of $\mathbf{u}$ as being the $\mathrm{z}$-axis for a coordinate system at $\mathbf{p}$. Stand at $\mathbf{p}+\mathbf{u}$ and look towards the “origin” $\mathbf{p}$. The counterclockwise direction in the ” $\mathrm{x}-\mathrm{y}$ plane” of this coordinate system will then determine the positive direction for an angle. See Figure 2.27. More precisely, choose vectors $\mathbf{u}{1}$ and $\mathbf{u}{2}$ so that $\left(\mathbf{u}{1}, \mathbf{u}{2}, \mathbf{u}\right)$ forms an orthonormal basis for $\mathbf{R}^{3}$ that induces the standard orientation. Then $\left(\mathbf{u}{1}, \mathbf{u}{2}\right)$ induces the desired orientation on the $\mathrm{x}-\mathrm{y}$ plane of the coordinate system from which “clockwise” and “counterclockwise” are determined. The rule can also be expressed in terms of the so-called “right-hand rule,” that is, if one lets the thumb of one’s right hand point in the direction of $\mathbf{u}$, then the curl of the fingers will specify the positive direction of angles. See Figure 2.27 again.

## 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