# 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
• 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|Sphere Model for a Convex Cone

One often chooses a representative for each open ray, in order to replace open rayswhich are infinite sets of points-by single points. A convenient way to do this is to normalize: that is, to choose the unit vector on each ray. Thus the one-sided directions in the space $X=\mathbb{R}^{n}$ are modeled as the points on the standard unit sphere $S_{X}=S_{n}={x \in X \mid|x|=1}$ in $X$. The set of unit vectors in a convex cone $C \subseteq X=\mathbb{R}^{n}$ will be called the sphere model for $C$. The subsets of the standard unit sphere $S_{X}$ that one gets in this way are precisely the geodesically convex subsets of $S_{X}$. A subset $T$ of $S_{X}$ is called geodesically convex if for each two different points $p, q$ of $T$ that are not antipodes $(p \neq-q)$, the shortest curve on $S_{X}$ that connects them is entirely contained in $T$. This curve is called the geodesic connecting these two points. Note that for two different points $p, q$ on $S_{X}$ that are not antipodes, there is a unique great circle on $S_{X}$ that contains them. A great circle on $S_{X}$ is a circle on $S_{X}$ with center the origin, that is, it is the intersection of the sphere $S_{X}$ with a two dimensional subspace of $X$. This great circle through $p$ and $q$ gives two curves on $S_{X}$ on this circle connecting the two points, a short one and a long one. The short one is the geodesic connecting these points. Example 1.4.3 (Sphere Model for a Convex Cone)

1. Figure $1.5$ above illustrates the sphere model for convex cones in dimension two. The convex cone $C$ is modeled by an arc.
2. Figure $1.6$ illustrates the sphere model for convex cones in dimension three, and it illustrates the concepts great circle and geodesic on $S_{X}$ for $X=\mathbb{R}^{3}$.

Two great circles are drawn. These model two convex cones that are planes through the origin. For the two marked points on one of these circles, the segment on this circle on the front of the sphere connecting the two points is their geodesic. Moreover, you see that two planes in $\mathbb{R}^{3}$ through the origin (and remember that a plane is a convex cone) are modeled in the sphere model for convex cones by two large circles on the sphere.

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

Often, we will be working with convex cones in $X=\mathbb{R}^{n}$ that lie above or on the horizontal coordinate hyperplane $x_{n}=0$. Then the unit vectors of these convex cones lie on the upper hemisphere $x_{1}^{2}+\cdots+x_{n}^{2}=1, x_{n} \geq 0$. Then the sphere model is called the hemisphere model. Example 1.4.4 (Hemisphere Model for a Convex Cone)

1. Figure $1.5$ above illustrates the hemisphere model for convex cones in dimension two. The convex cone $C$ is modeled by an arc on the upper half-circle.
2. Figure $1.8$ illustrates the hemisphere model for three convex cones $C$ in dimension three.

The models of these three convex cones $C$ are shaded regions on the upper hemisphere. These regions are indicated by the same letter as the convex cone that they model, by $C$. The number of points in the model of $C$ that lie on the circle that bounds the hemisphere is, 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|Sphere Model for a Convex Cone

1. 数字 $1.5$ 上面说明了二维凸雉的球体模型。凸雉 $C$ 由圆弧建模。
2. 数字 $1.6$ 说明了三维凸雉的球体模型，并说明了大圆和测地线的概念 $S_{X}$ 为了 $X=\mathbb{R}^{3}$. 画了两个大圆圈。它们模拟了两个凸雉，它们是通过原点的平面。对于其中一个圆上的两个 标决点，连接纹两个点的球体前面的这个圆上的线段是它们的测地线。此外，你看到有两架 飞机在 $\mathbb{R}^{3}$ 通过原点 (并记住平面是凸雉) 在球体模型中通过球体上的两个大圆来模拟凸 锥。

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

1. 数字 $1.5$ 上面说明了二维凸锥的半球模型。凸雉 $C$ 由上半圆上的弧线建模。
2. 数字 $1.8$ 说明了三个凸锥的半球模型 $C$ 在第三维度。 这三个凸雉的模型 $C$ 是上半球的阴影区域。这些区域由与它们建模的凸雉相同的字母表示， 由 $C$. 模型中的点数 $C$ 位于半球边界的圆上，从左到右依次是: 零、一、无限多。这意味着 凸雉的水平射线数 $C$ 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