# 线性代数网课代修|凸分析代写Convex analysis代考|EEEN422

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• 逼近论

## 线性代数作业代写linear algebra代考|Non-uniqueness Homogenization

We have already stated the fact that each convex set $A \subseteq X$ is the deconification of a suitable convex cone $C \subseteq X \times \mathbb{R}_{+}$. This convex cone $C$, a homogenization of $A$, is not unique. To give a precise description of the non-uniqueness we need the important concept recession direction.

Definition 1.5.4 Let $A \subseteq X=\mathbb{R}^{n}$ be a convex set. A recession vector of $A$ is a vector $c \in X$ such that $a+t c \in A$ for all $a \in A$ and all $t \geq 0$. This means, if $c \neq 0_{n}$, that one can travel in $A$ in the one-sided direction of $c$ without ever leaving $A$. So you can escape to infinity within $A$ in this direction. The recession cone $R_{A}$ of the convex set $A$ is the set in $X$ consisting of all recession vectors of $A$. The set $R_{A}$ is a convex cone. A recession direction of $A$ is a one-sided direction of $R_{A}$, that is, an open ray of the convex cone $R_{A}$.

Fxample 1.5.5 (Recession Cone (Geometric)) Figure $1.11$ illustrates the concept of recession cone for convex sets with boundary an ellipse, a parabola and a branch of a hyperbola respectively.

On the left, a closed convex set $A$ is drawn with boundary an ellipse. This set is bounded, there are no recession directions: $R_{A}=\left{0_{n}\right}$. In the middle, a closed convex set $A$ is drawn with boundary a parabola. This set is unbounded, there is a unique recession direction: $R_{A}$ is a closed ray. This recession direction is indicated by a dotted half-line: $c$ is a nonzero recession vector. On the right, a closed convex set $A$ is drawn with boundary a branch of a hyperbola. This set is unbounded, there are infinitely many recession directions: $R_{A}$ is the closed convex cone in the plane with legs the directions of the two asymptotes to the branch of the hyperbola that is the boundary of $A$.

## 线性代数作业代写linear algebra代考|Convex Sets: Visualization by Models

For visualization of convex sets $A$ in $X=\mathbb{R}^{n}$, we will make use of the ray model, the hemisphere model and the top-view model for a convex set.

Example 1.6.1 (Ray Model, Hemisphere Model and Top-View Model for a Convex Set) Figure $1.12$ illustrates the three models in the case of a convex set in the line $\mathbb{R}$.

A convex set $A \subseteq \mathbb{R}$ is drawn; $\mathbb{R}$ is identified with the horizontal axis in the plane $\mathbb{R}^{2}$; the set $A$ is lifted up to level 1; the open rays that start from the origin and that run through a point of this lifted up set form the ray model for $A$; the arc that consists of the intersections of these rays with the upper half-circle form the hemisphere model for $A$; the orthogonal projection of this arc onto the horizontal axis forms the top-view model for $A$.

The three models for visualization of convex sets are the combination of two constructions that have been presented in the previous two sections: homogenization of convex sets in $X$, and visualization of convex cones in $X \times \mathbb{R}=\mathbb{R}^{n+1}$ lying above the horizontal hyperplane $X \times{0}$ by means of the ray, hemisphere and top-view model for convex cones.

Ray Model for a Convex Set Each point $x \in X=\mathbb{R}^{n}$ can be represented by the open ray in $X \times \mathbb{R}{++}$that contains the point $(x, 1)$. Thus each subset of $X$ is modeled as a set of open rays in the open upper halfspace $X \times \mathbb{R}{++}$. This turns out to be specially fruitful for visualizing convex sets, if we want to understand their behavior at infinity. We will see this in Chap. 3. We emphasize that this is an important issue. Hemisphere Model for a Convex Set Consider the upper hemisphere $$\left{(x, \rho) \mid x \in X, \rho \in \mathbb{R}{+}, x{1}^{2}+\cdots+x_{n}^{2}+\rho^{2}=1\right}$$ in $X \times \mathbb{R}=\mathbb{R}^{n+1}$. Each point $x$ in $X=\mathbb{R}^{n}$ can be represented by a point in this hemisphere: the intersection of the hemisphere with the open ray that contains the point $(x, 1)$. Thus $x \in X$ is represented by the point $|(x, 1)|^{-1}(x, 1)$ on the hemisphere, where $|\cdot|$ denotes the Euclidean norm. Thus we get for each subset of $X$, and in particular for each convex set in $X$, a bijection with a subset of the hemisphere. This set has no points in common with the boundary of the hemisphere, which is a sphere in one dimension lower, in $\mathbb{R}^{n-1}$.

## 线性代数作业代写linear algebra代考|Non-uniqueness Homogenization

Fxample 1.5.5 (衰退雉（几何) ) 图1.11分别说明了边界为椭圆、抛物线和双曲线分支的 凸集的衰退雉的概念。 在左边，一个闭凸集 $A$ 以椭圆为边界绘制。这个集合是有界的，没有衰退方向： 个独特的衰退方向: $R_{A}$ 是闭合射线。这种衰退方向由一条虚线半线表示： $c$ 是一个非零衰 退向量。右边是闭凸集 $A$ 用边界画出一条双曲线的分支。这个集合是无界的，有无限多的衰 退方向： $R_{A}$ 是平面上的封闭凸锥，两条渐近线的方向指向作为边界的双曲线的分支 $A$.

## 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