# 线性代数网课代修|凸分析代写Convex analysis代考|SS2022

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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|Top-View Model for a Convex Cone

In fact, for a convex cone $C \subseteq X=\mathbb{R}^{n}$ that is contained in the halfspace $x_{n} \geq 0$, one can use an even simpler model, in one dimension lower-in $\mathbb{R}^{n-1}$ instead of in $\mathbb{R}^{n}$. Suppose we look at the hemisphere from high above it. Here we view the $n$-th coordinate axis of $X=\mathbb{R}^{n}$ as a vertical line. Then the hemisphere looks like the standard closed unit ball in dimension $n-1$, and the subset of the hemisphere that models $C$ looks like a subset of this ball. To be more precise, one can take the orthogonal projection of points on the hemisphere onto the hyperplane $x_{n}=0$, by setting the last coordinate equal to zero. Then the hyperplane $x_{n}=0$ in $X=\mathbb{R}^{n}$ can be identified with $\mathbb{R}^{n-1}$ by omitting the last coordinate 0 . This gives that the set of one-sided directions of the convex cone is modeled as a subset of the standard closed unit ball in $\mathbb{R}^{n-1}$, $$B_{\mathbb{R}^{n-1}}=B_{n-1}=\left{x \in \mathbb{R}^{n-1} \mid x_{1}^{2}+\cdots+x_{n-1}^{2} \leq 1\right} .$$ This subset will be called the top-view model for a convex cone. Example 1.4.5 (Top-View Model for a Convex Cone)

1. Figure $1.5$ above illustrates the top-view model for convex cones in dimension two. Here the convex cone $C$ is modeled by a closed line segment contained in the closed interval $[-1,1]$, the standard closed unit ball $B_{1}$ in $\mathbb{R}^{1}$.
2. Figure $1.9$ illustrates the top-view model for three convex cones $C$ in dimension three.

The models for these three convex cones $C$ are shaded regions in the standard closed unit disk $B_{2}$ in the plane $\mathbb{R}^{2}$, which are indicated by the same letter as the convex cones that they model, by $C$. All three convex cones that are modeled, are chosen in such a way that they contain the positive vertical coordinate axis ${(0,0, \rho) \mid \rho>0}$ in their interior. As a consequence, their top-view models contain in their interior the center of the disk. The arrows represent images under orthogonal projection on the plane $x_{3}=0$ of some geodesics that start at the top of the hemisphere. The number of points in the model of $C$ that lie on the circle are, from left to right: zero, one, infinitely many. This means that the number of horizontal rays of the convex cone $C$ is, from left to right: zero, one, infinitely many.

## 线性代数作业代写linear algebra代考|Homogenization: Definition

Now we are going to present homogenization of convex sets, the main ingredient of the unified approach to convex analysis that is used in this book. That is, we are going to show that every convex set in $X=\mathbb{R}^{n}$-and we recall that this is our basic object of study – can be described by a convex cone in a space one dimension higher, $X \times \mathbb{R}=\mathbb{R}^{n+1}$, that lies entirely on or above the horizontal coordinate hyperplane $x_{n+1}=0$, that is, it lies in the closed upper halfspace $x_{n+1} \geq 0$.

Example 1.5.1 (Homogenization of a Bounded Convex Set) Figure $1.10$ illustrates the idea of homogenization of convex sets for a bounded convex set $A$ in the plane $\mathbb{R}^{2} .$ This crucial picture can be kept in mind throughout this book! The plane $\mathbb{R}^{2}$ is drawn as the floor, $\mathbb{R}^{2} \times{0}$, in three dimensional space $\mathbb{R}^{3}$. The convex set $A$ is drawn as lying on the floor, as the set $A \times{0}$, and this copy is indicated by the same letter as the set itself, by $A$. This set is lifted upward in vertical direction to level 1 , giving the set $A \times{1}$. Then the union of all open rays that start at the origin and that run through a point of the set $A \times{1}$ is taken. This union is defined to be the homogenization or conification $C=c(A)$ of $A$. Then $A$ is called the (convex set) dehomogenization or deconification of $C$.

There is no loss of information if we go from $A$ to $C$ : we can recover $A$ from $C$ as follows. Intersect $C$ with the horizontal plane at level 1. This gives the set $A \times{1}$. Then drop this set down on the floor in vertical direction. This gives the set $A$, viewed as lying on the floor.

## 线性代数作业代写linear algebra代考|Top-View Model for a Convex Cone

1. 数字 $1.5$ 上面说明了二维凸锥的顶视图模型。这里是凸锥 $C$ 由包含在闭合区间中的闭合 线段建模 $[-1,1]$ ，标准封闭单元球 $B_{1}$ 在 $\mathbb{R}^{1}$.
2. 数字 $1.9$ 说明了三个凸锥的顶视图模型 $C$ 在第三维度。 这三个凸雉的模型 $C$ 是标准封闭单位圆盘中的阴影区域 $B_{2}$ 在飞机上 $\mathbb{R}^{2}$ ，它们由与它们建模 的凸雉相同的字母表示，由 $C$. 被建模的所有三个凸锥都以包含正垂直坐标轴的方式选择 $(0,0, \rho) \mid \rho>0$ 在他们的内部。因此，他们的顶视图模型在其内部包含磁盘的中心。箭头 表示平面上正交投影下的图像 $x_{3}=0$ 从半球顶部开始的一些测地线。模型中的点数 $C$ 位于 圆上的从左到右依次为：零、一、无限多。这意味着凸锥的水平射线数 $C$ 是，从左到右： 零，一，无限多。

## 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