线性代数网课代修|凸分析代写Convex analysis代考|CS531

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线性代数作业代写linear algebra代考|Visualization by Models

The aim of this section is to present three models for convex cones $C \subseteq X=\mathbb{R}^{n}$ that lie in the upper halfspace $x_{n} \geq 0$ : the ray model, the hemisphere model and the top-view model.

Example 1.4.1 (Ray, Hemisphere and Top-View Model for a Convex Cone) Figure $1.5$ illustrates the three models in one picture for a convex cone $C$ in the plane $\mathbb{R}^{2}$ that lies above or on the horizontal axis.

The ray model visualizes this convex cone $C \subseteq \mathbb{R}^{2}$ by viewing $C \backslash\left{0_{2}\right}$ as a collection of open rays, and then considering the set whose elements are these rays; the hemisphere model visualizes $C$ by means of an arc: the intersection of $C$ with the upper half-circle $x_{1}^{2}+x_{2}^{2}=1, x_{2} \geq 0$; the top-view model visualizes $C$ by looking from high above at the arc, or, to be precise, by taking the interval in $[-1,+1]$ that is the orthogonal projection of the arc onto the horizontal axis.

The three models are all useful for visualization in low dimensions. Fortunately, most, maybe all, properties of convex cones in $\mathbb{R}^{n}$ for general $n$ can be understood by looking at such low dimensional examples. Therefore, making pictures in your head or with pencil and paper by means of one of these models is of great help to understand properties of convex cones-and so to understand the entire convex analysis, as we will see.

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

The role of nonzero elements of a convex cone $C \subseteq X=\mathbb{R}^{n}$ is usually to describe one-sided directions. Then, two nonzero elements of $C$ that differ by a positive scalar multiple, $a$ and $\rho a$ with $a \in \mathbb{R}^{n}$ and $\rho>0$, are considered equivalent. The equivalence classes of $C \backslash\left{0_{n}\right}$ are open rays of $C$, sets of all positive multiples of a nonzero element. So, what often only matters about a convex cone $C$ is its set of open rays, as this describes a set of one-sided directions. The set of open rays of $C$ will be called the ray model for the convex cone $C$.

The ray model is the most simple description of the one-sided directions of $C$ from a mathematical point of view, but it might be seen as inconvenient that a onesided direction is modeled by an infinite set. Moreover, it requires some preparations such as the definition of distance between two open rays: this is needed in order to define convergence of a sequence of rays. Suppose we have two open rays, ${\rho v \mid \rho>$ $0}$ and ${\rho w \mid \rho>0}$, where $v, w \in X$ are unit vectors, that is, their lengths or Euclidean norms, $|v|=\left(v_{1}^{2}+\cdots+v_{n}^{2}\right)^{\frac{1}{2}}$ and $|w|=\left(w_{1}^{2}+\cdots+w_{n}^{2}\right)^{\frac{1}{2}}$, are both equal to 1 . Then the distance between these two rays can be taken to be the angle $\varphi \in[0, \pi]$ between the rays, which is defined by $\cos \varphi=v \cdot w=v_{1} w_{1}+\cdots+v_{n} w_{n}$, the dot product of $v$ and $w$. To be precise about the concept of distance, the concept of metric space would be needed; however, we will not consider this concept.

Example 1.4.2 (Ray Model for a Convex Cone) Figure $1.5$ above illustrates the ray model for convex cones in dimension two. A number of rays of the convex cone $C$ are drawn.

线性代数作业代写linear algebra代考|Visualization by Models

$x_{1}^{2}+x_{2}^{2}=1, x_{2} \geq 0$; 顶视图模型可视化 $C$ 通过从高处看弧线，或者更准确地说，通过在 $[-1,+1]$ 即圆弧在水平轴上的正交投影。

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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