# 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
• 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 Revisited

The last section described what might be called the geometric approach to defining motions in $\mathbf{R}^{3}$. Some of the computations got rather complicated. The power of frames comes from their ability to define a motion $\mathrm{M}$ in terms of an orthonormal basis, which is typically easier to define than the rotations and reflections that might describe $M$ if we were to use the approach from the last section. We saw some of this in Section $2.2 .8$, but it is especially going to pay off here. As our first example we redo Example 2.5.1.4 2.5.2.1. Example. To find the rotation $\mathrm{R}$ that rotates the plane $\mathbf{X}$ defined by $$\frac{3}{2} x+3 y+z=0$$ to the $x-y$ plane. Solution. We use the same notation as in Example 2.5.1.4. See Figure 2.29. Applying the Gram-Schmidt algorithm to the basis $\mathbf{A}(2,0,-3)$ and $\mathbf{B}(0,1,-3)$ for $\mathbf{X}$ gives us an orthonormal basis $$\mathbf{u}{1}=\frac{1}{\sqrt{13}}(2,0,-3), \quad \text { and } \quad \mathbf{u}{2}=\frac{1}{7 \sqrt{13}}(-18,13,-12)$$ The equation for $\mathbf{X}$ tells us that $\mathbf{n}=(3 / 2,3,1)$ is a normal vector for the plane. Let $$\mathbf{u}_{3}=\frac{\mathbf{n}}{|\mathbf{n}|}=\frac{1}{7}(3,6,2),$$ and consider the frame $\mathrm{F}=\left(\mathbf{u}{1}, \mathbf{u}{2}, \mathbf{u}_{3}\right)$. The rotation $\mathrm{R}$ defined by $\mathrm{F}^{-1}$ then solves the problem. The matrix for $\mathrm{R}$ is the same one as we got before, namely, $$\left(\begin{array}{ccc} \frac{2}{\sqrt{13}} & -\frac{18}{7 \sqrt{13}} & \frac{3}{7} \ 0 & \frac{\sqrt{13}}{7} & \frac{6}{7} \ -\frac{3}{\sqrt{13}} & -\frac{12}{7 \sqrt{13}} & \frac{2}{7} \end{array}\right)$$ Actually, the fact that we got the same answer is accidental since the problem is underconstrained and there are many rotations that rotate $\mathbf{X}$ to the $\mathrm{x}-\mathrm{y}$ plane.

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

The last chapter outlined some of the basic elements of affine geometry. This chapter looks at projective geometry. Some general references that look at the subject in more detail than we are able to here are [Ayre67], [Gans69], and [PenP86].

Like in the last chapter, we shall start with dimension two (Sections 3.2-3.4) and only get to higher dimensions in Section 3.5. In order to motivate the transition from affine geometry to projective geometry we begin by studying projective transformations in affine space. Section $3.2$ starts off by looking at central projections and leads up to a definition of a projective transformation of the plane. We shall quickly see that, in contrast to affine geometry, we have to deal with certain exceptional cases that make the statement of definitions and theorems rather awkward. Mathematicians do not like having to deal with results on a case-by-case basis. Furthermore, the existence of special cases often is a sign that one does not have a complete understanding of what is going on and that there is still some underlying general principle left to be discovered. In fact, it will become clear that Euclidean affine space is not the appropriate space to look at when one wants to study projective transformations and that one should really look at a larger space called projective space. This will allow us to deal with our new geometric problems in a uniform way.

Projective space itself can be introduced in different ways. One can start with a synthetic and axiomatic point of view or one using coordinates. Lack of space prevents us from discussing both approaches and so we choose the latter because it is more practical. In Section $3.3$ we introduce homogeneous coordinates after a new look at points and lines that motivates the point of view that projective space is a natural coordinate system extension of Euclidean space. This leads to a definition and discussion of the projective plane $\mathbf{P}^{7}$ in Sertion 3.4. Snme of its important analytis properties are described in Section 3.4.1. Sections $3.4 .2$ and 3.4.3 define projective transformations of $\mathbf{P}^{2}$ and show how affine transformations are just special cases if one uses homogeneous coordinates. We then generalize to higher dimensions in Section 3.5. The important special case of 3-dimensional projective transformations is considered in Section 3.5.1. Next we study conics (Sections $3.6$ and 3.6.1) and quadric surfaces (Section 3.7).We finish the chapter with several special topics. Section $3.8$ discusses a generalization of the usual central projection. Section $3.9$ describes the beautiful theorem of Pascal and some applications. The last topic of the chapter is the stereographic projection. Section $3.10$ describes some of its main properties.

## 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