线性代数网课代修|凸分析代写Convex analysis代考|ESE605

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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|Primal Description of the Convex Hull

The convex hull can be described explicitly as follows. We need a definition. Definition 1.7.6 A convex combination or weighted average of a finite set $S=$ $\left{s_{1}, \ldots, s_{k}\right} \subseteq X=\mathbb{R}^{n}$ is a nonnegative linear combination of the elements of $S$ with sum of the coordinates equal to 1 , $$\rho_{1} s_{1}+\cdots+\rho_{k} s_{k}$$ with $\rho_{i} \in \mathbb{R}{+}, 1 \leq i \leq k$ and $\rho{1}+\cdots+\rho_{k}=1$. Example 1.7.7 (Convex Combination) Let $a, b, c$ be three points in $\mathbb{R}^{2}$ that do not lie on one line. Then the set of convex combinations of $a, b, c$ is the triangle with vertices $a, b, c$.

Proposition 1.7.8 The convex hull of a set $S \subseteq X=\mathbb{R}^{n}$ consists of the convex combinations of finite subsets of $S$.

In particular, a convex set is closed under taking convex combinations of a finite set of points.

This could be called a primal description – or a description from the inside-of a convex set. Each choice of a finite subset of the convex set gives an approximation from the inside: the set of all convex combinations of this subset. Figure $1.17$ illustrates the primal description of a convex set.

Proposition $1.7 .8$ will be proved by the homogenization method. This is the first of many proofs of results for convex sets that will be given in this way. Therefore, we make here explicit how all these proofs will be organized.

1. We begin by stating a version of the result for convex cones, that is, for homogeneous convex sets. This version is obtained by conification (or homogenization) of the original statement, which is for convex sets. For brevity we only display the outcome, which is a statement for convex cones. So we do not display how we got this statement for convex cones from the original statement for convex sets.
2. Then we prove this convex cone version, working with convex cones.
3. Finally, we translate back by deconification (or dehomogenization).

线性代数作业代写linear algebra代考|Definition of Three Operations

The first operation is the Cartesian product. If we have two convex sets $A \subseteq Y=$ $\mathbb{R}^{m}, B \subseteq X=\mathbb{R}^{n}$, then their Cartesian product is the set $$A \times B={(a, b) \mid a \in A, b \in B}$$ in $Y \times X=\mathbb{R}^{m} \times \mathbb{R}^{n}=\mathbb{R}^{m+n}$. This is a convex set.

The second operation is the image of a convex set under a linear function. Let $M$ be an $m \times n$-matrix. This matrix determines a linear function $\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ by the recipe $x \mapsto M x$. For each convex set $A \subseteq \mathbb{R}^{n}$, its image under $M$ is the set $$M A={M a \mid a \in A}$$ in $\mathbb{R}^{m}$. This is a convex set. Example 1.8.1 (Image Convex Set Under Linear Function) Let $M$ be a $2 \times 2$-matrix, neither of whose two columns is a scalar multiple of the other column.

1. Let $A$ be a rectangle in the plane $\mathbb{R}^{2}$ whose sides are parallel to the two coordinate axes, $a \leq x<b, c \leq y<d$ for constants $a<b, c<d$. Then $M A$ is the parallelogram with vertices $M\left[\begin{array}{l}a \ c\end{array}\right], M\left[\begin{array}{l}b \ c\end{array}\right], M\left[\begin{array}{l}a \ d\end{array}\right], M\left[\begin{array}{l}b \ d\end{array}\right]$.
2. An arbitrary convex set $A \subseteq \mathbb{R}^{2}$ can be approximated by the disjoint union of many small rectangles, and then by the result above one gets an approximation of $M A$ as a disjoint union of parallelograms.

The third operation is the inverse image of a convex set under a linear function. Let $M$ be as above. For each convex set $B \subseteq \mathbb{R}^{m}$, its inverse image under $M$ is the set $$B M=\left{a \in \mathbb{R}^{n} \mid M a \in B\right}$$ in $\mathbb{R}^{n}$. This is a convex set. These three operations also preserve the convex cone property.

线性代数作业代写linear algebra代考|Primal Description of the Convex Hull

1. 我们首先说明凸雉的结果的一个版本，即齐次凸集。这个版本是通过对凸集的原始陈 述的conification (或同质化) 获得的。为简洁起见，我们只显示结果，这是对凸锥的 陈述。所以我们没有展示我们是如何从凸集的原始陈述中得到这个关于凸锥的陈述 的。
2. 然后我们证明这个凸雉版本，使用凸雉。
3. 最后，我们通过去雉化 (或去均质化) 翻译回来。

线性代数作业代写linear algebra代考|Definition of Three Operations

1. 让 $A$ 是平面中的一个矩形 $\mathbb{R}^{2}$ 其边平行于两个坐标轴， $a \leq x<b, c \leq y<d$ 对于常数 $a<b, c<d$. 然后 $M A$ 是有顶点的平行四边形 $M[a c], M[b c], M[a d], M[b d]$.
2. 任意凸集 $A \subseteq \mathbb{R}^{2}$ 可以由许多小矩形的不相交并集来近似，然后由上面的结果得到一个 近似值 $M A$ 作为平行四边形的不相交并集。 第三个操作是凸集在线性函数下的逆像。让 $M$ 如上。对于每个凸集 $B \subseteq \mathbb{R}^{m}$ ，它的逆像下 $M$ 是集合 B $M=\backslash$ left ${a \backslash$ in $\backslash$ mathbb ${R} \wedge{n} \backslash$ mid $M a \backslash$ in B\right } } 在 $\mathbb{R}^{n}$. 这是一个凸集。 这三个操作还保留了凸锥属性。

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