# 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|The Three Golden Convex Cones

The most celebrated basic examples of convex sets are the following three ‘golden’ convex cones.

• The first orthant $\mathbb{R}_{+}^{n}$, is the set of points in $\mathbb{R}^{n}$ with nonnegative coordinates.
• The Lorentz cone $L^{n}$, also called the ice cream cone, is the epigraph of the Euclidean norm on $\mathbb{R}^{n}$, that is, the region in $\mathbb{R}^{n+1}$ above or on the graph of the Euclidean norm,
$$\left{(x, y) \mid x \in \mathbb{R}^{n}, y \in \mathbb{R}, \quad y \geq\left(x_{1}^{2}+\cdots+x_{n}^{2}\right)^{\frac{1}{2}}\right} .$$
• The positive semidefinite cone $\mathbb{S}_{+}^{n}$, is the set of positive semi-definite $n \times n$ matrices in the vector space of symmetric $n \times n$-matrices $\mathbb{S}^{n}$ (a symmetric $n \times n$ matrix $M$ is called positive semidefinite if $x^{\top} M x \geq 0$ for all $x \in \mathbb{R}^{n}$ ).

Note that in the third example we do not work in $\mathbb{R}^{n}$, but in the space $\mathbb{S}^{n}$ of symmetric $n \times n$-matrices. This can be justified by identifying this space with $\mathbb{R}^{m}$ where $m=\frac{1}{2} n(n+1)$ by stacking symmetric matrices, that is, by taking the entries on or above the main diagonal of a symmetric matrix and then writing them as one long column vector, taking the columns of the matrix from left to right and writing these entries for each column from top to bottom.

The convex cone $\mathbb{S}_{+}^{n}$ models in terms of linear algebra (and so relatively simply) a solution set of a system of nonlinear inequalities (that is, a complicated set).

Example 1.7.1 (Positive Semi-definite Cone) If we identify a symmetric $2 \times 2$ matrix $B$ with the vector in $\mathbb{R}^{3}$ with coordinates $b_{11}, b_{22}, b_{12}$ respectively, then the convex cone $\mathbb{S}{+}^{2}$ gets identified with the convex set $x{1} x_{2}-x_{3}^{2} \geq 0, x_{1} \geq 0, x_{2} \geq 0$.

## 线性代数作业代写linear algebra代考|Convex Hull and Conic Hull

Here is a source of many basic examples of convex sets.
Definition 1.7.2 To each set $S \subseteq X=\mathbb{R}^{n}$, one can add points from $X$ in a minimal way in order to make a convex set $\operatorname{co}(S) \subseteq X$, the convex hull of $S$. That is, $\operatorname{co}(S)$ is the smallest convex set containing $S$. It is the intersection of all convex sets containing $S$.
Example 1.7.3 Figure $1.15$ illustrates the concept of convex hull.
From left to right: the first two sets are already convex, so they are equal to their convex hull; for the other convex sets, taking their convex hull means respectively filling a dent, filling a hole, and making from a set consisting of two pieces a set consisting of one piece, without dents.
Here is a similar source of examples of convex cones.
Definition 1.7.4 To each set $S \subseteq X$, one can adjoin to $S \cup\left{0_{X}\right}$ points from $X$ in a minimal way in order to make a convex cone cone( $S)$, the conic hull of $S$. That is, cone $(S)$ is the smallest convex cone containing $S$ and the origin. It is the intersection of all convex cones containing $S$ and the origin.
Example 1.7.5 (Conic Hull)

1. Figure $1.16$ illustrates the concept of conic hull.
This picture makes clear that a lot of structure can get lost if we pass from a set to its conic hull.
2. The conification of a convex set $A \subset \mathbb{R}^{n}$ is essentially the conic hull of $A \times{1}$ :
$$c(A)=\operatorname{cone}(A \times{1}) \backslash\left{0_{X \times \mathbb{R}}\right}$$

## 线性代数作业代写linear algebra代考|The Three Golden Convex Cones

• 第一颗 $\mathbb{R}_{+}^{n}$, 是点的集合 $\mathbb{R}^{n}$ 具有非负坐标。
• 洛伦兹锥 $L^{n}$, 也称为冰淇淋蛋筒，是欧几里得范数的题词 $\mathbb{R}^{n}$, 即区域在 $\mathbb{R}^{n+1}$ 在欧几里 得范数之上或之上，
left $\left{(x, y) \backslash \operatorname{mid} x \backslash\right.$ in $\backslash$ mathbb ${R} \wedge{n}, y \backslash$ in $\backslash$ mathbb ${R}, \backslash$ quad $y \backslash$ geq $\backslash$ left $\left(x_{-}{1} \wedge{2}+\backslash\right.$ cdots $+x_{-}{n} \wedge$
• 正半定锥 $\mathbb{S}{+}^{n}$, 是正半定的集合 $n \times n$ 对称向量空间中的矩阵 $n \times n$-矩阵 $\mathbb{S}^{n}$ (一个对称 的 $n \times n$ 矩阵 $M$ 称为半正定如果 $x^{\top} M x \geq 0$ 对所有人 $x \in \mathbb{R}^{n}$ ). 请注意，在第三个示例中，我们不使用 $\mathbb{R}^{n}$, 但在空间 $\mathbb{S}^{n}$ 对称的 $n \times n$-矩阵。这可以通过识 别这个空间来证明 $\mathbb{R}^{m}$ 在哪里 $m=\frac{1}{2} n(n+1)$ 通过堆㿿对称矩阵，即，取对称矩阵主对角 线之上或之上的元素，然后将它们写为一个长列向量，从左到右取矩阵的列，并为每一列写 下这些元素从上到下。 凸雉 $\mathbb{S}{+}^{n}$ 根据线性代数（因此相对简单）建模非线性不等式系统的解集（即复杂集）。
示例 1.7.1 (正半定雉) 如果我们确定一个对称 $2 \times 2$ 矩阵 $B$ 向量在 $\mathbb{R}^{3}$ 带坐标 $b_{11}, b_{22}, b_{12}$ 分 别是凸锥 $S+{ }^{2}$ 被识别为凸集 $x 1 x_{2}-x_{3}^{2} \geq 0, x_{1} \geq 0, x_{2} \geq 0$.

## 线性代数作业代写linear algebra代考|Convex Hull and Conic Hull

$S)$, 的圆雉壳 $S$. 也就是锥 $(S)$ 是包含的最小凸锥 $S$ 和起源。它是所有凸锥的交集，包含 $S$ 和起 源。

1. 数字1.16说明了圆雉壳的概念。
这张照片清楚地表明，如果我们从一个集合传递到它的圆雉形外壳，很多结构可能会
丢失。
2. 凸集的conification $A \subset \mathbb{R}^{n}$ 本质上是圆锥壳 $A \times 1:$

# 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