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

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## 线性代数作业代写linear algebra代考|Fair Bargains

The aim of this section is to give an example of how a convex set can arise in a natural way. You can skip this section, and similar first sections in later chapters, if you want to come to the point immediately.

John Nash has solved the problem of finding a fair outcome of a bargaining opportunity (see [1]). This problem does not involve convex sets, but its solution requires convex sets and their properties.

Here is how I learned about Nash bargaining. Once I was puzzled by the outcome of bargaining between twó students, exchanging a textboook and sunglassés. Thé prices of book and glasses were comparable-the sunglasses being slightly more expensive. The bargain that was struck was that the student who owned the most expensive item, the sunglasses, paid an additional amount of money. What could be the reason for this? Maybe this student had poor bargaining skills? Someone told me that the problem of what is a fair bargain in the case of equal bargaining skills had been considered by Nash. The study of this theory, Nash bargaining, was an eye-opener: it explained that the bargain I had witnessed and that initially surprised me, was a fair one.

Nash bargaining will be explained by means of a simple example. Consider Alice and Bob. They possess certain goods and these have known utilities ( $\approx$ pleasure) to them. Alice has a bat and a box, Bob has a ball. The bat has utility 1 to Alice and utility 2 to Bob; the box has utility 2 to Alice and utility 2 to Bob; the ball has utility 4 to Alice and utility 1 to Bob.

If Alice and Bob would come to the bargaining agreement to exchange all their goods, they would both be better off: their total utility would increase-for Alice from 3 to 4 and for Bob from 1 to 4 . Bob would profit much more than Alice from this bargain. So this bargain does not seem fair.

Now suppose that it is possible to exchange goods with a certain probability. Then we could modify this bargain by agreeing that Alice will give the box to Bob with probability 1/2. For example, a coin could be tossed: if heads comes up, then Alice should give the box to Bob, otherwise Alice keeps her box. This bargain increases expected utility for Alice from 3 to 5 and for Bob from 1 to 3 . This looks more fair. More is true: we will see that this is the unique fair bargain in this situation.

## 线性代数作业代写linear algebra代考|Special Case of Subspaces

The aim of this section is to give some initial insight into the contents of Chaps. $1-4$ of this book. The aim of these chapters is to present the properties of convex sets and now we display the specializations of some of these properties from convex sets to special convex sets, subspaces. A subspace of $\mathbb{R}^{n}$ is a subset $L$ of $\mathbb{R}^{n}$ that is closed under sums and scalar multiples, $x, y \in L, \alpha, \beta \in \mathbb{R} \Rightarrow \alpha x+\beta y \in L . \mathrm{A}$ subspace can be viewed as the central concept from linear algebra: it is the solution set of a finite system of homogeneous linear equations in $n$ variables. A subspace is a convex cone and so, in particular, it is a convex set. The specializations of the results from Chaps. $1-4$ from convex sets to subspaces are standard results from linear algebra.

Chapter 1 A subspace of $X=\mathbb{R}^{n}$ is defined to be a nonempty subset $L \subseteq X$ that is closed under linear combinations of two elements: $$a, b \in L, \rho, \sigma \in \mathbb{R} \Rightarrow \rho a+\sigma b \in L .$$ Example 1.3.1 (False Equations and Homogenization) Now we illustrate homogenization of convex sets by means of a simple high school example. Before the use of the language of sets in high school, the following three cases were distinguished in some school books, for a system of two linear equations in two variables: \begin{aligned} &a_{11} x_{1}+a_{12} x_{2}=b_{1}, \ &a_{21} x_{1}+a_{22} x_{2}=b_{2} . \end{aligned}

## 线性代数作业代写linear algebra代考|Fair Bargains

John Nash 解决了寻找谈判机会的公平结果的问题（参见 [1]）。这个问题不涉及凸集，但它的解决方案需要凸集及其属性。

## 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