# 线性代数网课代修|同调代数代写homological algebra代考|MATH464

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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|Categories and Functors

Let us now pass from the concrete to the abstract. Categories are the context for discussing general properties of systems such as groups, rings, modules, sets, or topological spaces, in tandem with their respective transformations: homomorphisms, functions, or continuous maps.

There are well-known set-theoretic “paradoxes” showing that contradictions arise if we are not careful about how the undefined terms set and element are used. For example, Russell’s paradox gives a contradiction arising from regarding every collection as a set. Define a Russell set to be a set $S$ that is not a member of itself; that is, $S \notin S$, and define $R$ to be the collection of all Russell sets. Either $R$ is a Russell set or it is not a Russell set. If $R$ is a Russell set, then $R \notin R$, by definition. But all Russell sets lie in the collection of all Russell sets, namely, $R$; that is, $R \in R$, a contradiction. On the other hand, if $R$ is not a Russell set, then $R$ does not lie in the collection of all Russell sets; that is, $R \notin R$. But now $R$ satisfies the criterion for being a Russell set, another contradiction. We conclude that some conditions are needed to determine which collections are allowed to be sets. Such conditions are given in the Zermelo-Fraenkel axioms for set theory, specifically, by the axiom of comprehension; the collection $R$ is not a set, and this resolves the Russell paradox. Another approach to resolving this paradox involves restrictions on the membership relation: some say that $x \in x$ is not a well-formed formula; others say that $x \in x$ is well-formed, but it is always false.

Let us give a bit more detail. The Zermelo-Fraenkel axioms have primitive terms class and $\in$ and rules for constructing classes, as well as for constructing certain special classes, called sets. For example, finite classes and the natural numbers $\mathbb{N}$ are assumed to be sets. A class is called small if it has a cardinal number, and it is a theorem that a class is a set if and only if it is small; a class that is not a set is called a proper class.

## 线性代数作业代写linear algebra代考|Singular Homology

In the first section, we defined homology groups $H_{n}(X)$ for every finite simplicial complex $X$; we are now going to generalize this construction so that it applies to all topological spaces. Once this is done, we shall see that each $H_{n}$ is actually a functor Top $\rightarrow \mathbf{A} \mathbf{b}^{10}$ The reader will see that the construction has two parts: a topological half and an algebraic half.

Definition. Recall that Hilbert space is the set $\mathcal{H}$ of all sequences $\left(x_{i}\right)$, where $x_{i} \in \mathbb{R}$ for all $i \geq 0$, such that $\sum_{i=0}^{\infty} x_{i}^{2}<\infty$. Euclidean space $\mathbb{R}^{n}$ is the subset of $\mathcal{H}$ consisting of all sequences of the form $\left(x_{0}, \ldots, x_{n-1}, 0, \ldots\right)$ with $x_{i}=0$ for all $i \geq n$.

We begin by generalizing the notion of $n$-simplex, where a 0 -simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle (with interior), a 3-simplex is a (solid) tetrahedron, and so forth. Here is the precise definition. Definition. The standard $n$-simplex is the set of all (convex) combinations $$\Delta^{n}=\left[e_{0}, \ldots, e_{n}\right]=\left{t_{0} e_{0}+\cdots+t_{n} e_{n}: t_{i} \geq 0 \text { and } \sum_{i=0}^{n} t_{i}=1\right},$$ where $e_{i}$ denotes the sequence in $\mathcal{H}$ having 1 in the $i$ th coordinate and 0 everywhere else. We may also write $t_{0} e_{0}+\cdots+t_{n} e_{n}$ as the vector $\left(t_{0}, \ldots, t_{n}\right)$ in $\mathbb{R}^{n+1} \subseteq \mathcal{H}$. The ith vertex of $\Delta^{n}$ is $e_{i}$; the jth faces of $\Delta^{n}$, for $0 \leq j \leq n$, are the convex combinations of $j$ of its vertices.

## 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