# 线性代数网课代修|凸分析代写Convex analysis代考|MAC0084

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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|Preservation of Closedness and Properness

We make, for later purposes, the following observations for the two nice properties that convex sets can have, closedness and properness. The question is whether these properties are preserved under the three operations on convex sets that have been defined above. Closedness of convex sets is preserved by Cartesian products, by inverse images under linear functions, but not by images under linear functions.
Example 1.8.2 (Closedness Not Always Preserved Under Linear Image) Figure $1.19$ illustrates that the image of a closed convex set under a linear function is not closed.

The shaded region, the solution set of $x y \leq 1, x \geq 0$, is a closed convex set, but its orthogonal projection onto the $x$-axis is the open half-line $x>0, y=0$. It is not closed: it does not contain the origin.

A sufficient condition for preservation of closedness of a convex set $A \subseteq \mathbb{R}^{n}$ by the image under a linear function $\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ determined by an $m \times n$-matrix $M$ is that the kernel of $M$ (the solution set of $M x=0$ ) and the recession cone $R_{A}$ of $A$ have only the origin in common, as we will see in Chap. 3. Because of this, we will sometimes work with open convex sets instead of with closed sets: openness of convex sets is preserved by the image of a linear function $\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ determined by an $m \times n$-matrix $M$ provided this linear function is surjective, that is, provided the rows of $M$ are linearly independent. Properness of convex sets is preserved by Cartesian products, by images under linear functions, but not by inverse images under linear functions.

For a finite set of points in $X=\mathbb{R}^{n}$, we will give a concept of ‘points in the middle’.
Theorem 1.9.1 (Radon’s Theorem) A finite set $S \subset X=\mathbb{R}^{n}$ of at least $n+$ 2 points can be partitioned into two disjoint sets $B$ (‘blue’) and $R$ (‘red’) whose convex hulls intersect,
$$\operatorname{co}(B) \cap \operatorname{co}(R) \neq \emptyset .$$
Figure $1.20$ illustrates Radon’s theorem for four specific points in the plane and for five specific points in three-dimensional space.

A common point of the convex hulls of $B$ and $R$ is called a Radon point of $S$. It can be considered as a point in the middle of $S$.

1. Three points on the line $\mathbb{R}$ can be ordered $s_{1} \leq s_{2} \leq s_{3}$ and then their unique Radon point is $s_{2}$.
2. Four points in the plane $\mathbb{R}^{2}$ have either one point lying in the triangle formed by the other three points or they can be partitioned in two pairs of two points such that the two closed line segments with endpoints the points of a pair intersect. In both cases, there is a unique Radon point. In the first case, it is the one of these four points that lies in the triangle formed by the other three points. In the second case, it is the intersection of the two closed line segments.
Proof of Radon’s Theorem We use the homogenization method.
3. Convex cone version. We consider the following statement: A finite set $S \subset \mathbb{R}^{n-1} \times(0,+\infty)$-so each element of $S$ has positive last coordinate-of more than $n$ elements can be partitioned into two disjoint sets $B$ and $R$ whose conic hulls intersect in a nonzero point,
$$\operatorname{cone}(B) \cap \operatorname{cone}(R) \neq\left{0_{n}\right}$$

## 线性代数作业代写linear algebra代考|Preservation of Closedness and Properness

$$\operatorname{co}(B) \cap \operatorname{co}(R) \neq \emptyset .$$

1. 线上的三个点 $\mathbb{R}$ 可订购 $s_{1} \leq s_{2} \leq s_{3}$ 然后他们独特的氡点是 $s_{2}$.
2. 平面上的四个点 $\mathbb{R}^{2}$ 有一个点位于由其他三个点形成的三角形中，或者它们可以划分为 两对两点，使得两个闭合线段的端点与一对点相交。在这两种情况下，都有一个独特 的氡点。在第一种情况下，这四个点中的一个位于由其他三个点形成的三角形中。在 第二种情况下，它是两条闭合线段的交点。
3. 凸雉版本。我们考虑以下陈述: 有限集 $S \subset \mathbb{R}^{n-1} \times(0,+\infty)$ – 所以每个元素 $S$ 具有大 于的正最后坐标 $n$ 元素可以分成两个不相交的集合 $B$ 和 $R$ 其圆雉壳相交于一个非零点，

# 计量经济学代写

## 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