线性代数网课代修|同调代数代写homological algebra代考|MA3204

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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|Categorical Constructions

Imagine a set theory whose primitive terms, instead of set and element, are set and function. How could we define bijection, cartesian product, union, and intersection? Category Theory forces us to think in this way, for functors do not recognize elements. One nice aspect of thinking categorically is that we can see unexpected analogies; for example, we shall soon see that disjoint union in Sets, direct sum in ${ }_{R}$ Mod, and tensor product in ComRings are special cases of the same categorical notion. We now set ourselves the task of describing various constructions in Sets or in $\mathbf{A b}$ in such a way that they make sense in arbitrary categories.

Let us begin by investigating the notion of disjoint union of subsets. Two subsets $A$ and $B$ of a set can be forced to be disjoint. Consider the cartesian product $(A \cup B) \times{1,2}$ and its subsets $A^{\prime}=A \times{1}$ and $B^{\prime}=B \times{2}$. It is plain that $A^{\prime} \cap B^{\prime}=\varnothing$, for a point in the intersection would have coordinates $(a, 1)=(b, 2)$, which cannot be, for their second coordinates are not equal. We call $A^{\prime} \cup B^{\prime}$ the disjoint union of $A$ and $B$, and we note that it comes equipped with two functions, namely, $\alpha: A \rightarrow A^{\prime} \cup B^{\prime}$ and $\beta: B \rightarrow A^{\prime} \cup B^{\prime}$, defined by $\alpha: a \mapsto(a, 1)$ and $\beta: b \mapsto(b, 2)$. Denote $A^{\prime} \cup B^{\prime}$ by $A \cup B$.

Given functions $f: A \rightarrow X$ and $g: B \rightarrow X$, for some set $X$, there is a unique function $\theta: A \sqcup B \rightarrow X$ that extends both $f$ and $g$; namely, $$\theta(u)= \begin{cases}f(a) & \text { if } u=(a, 1) \in A^{\prime} \ g(b) & \text { if } u=(b, 2) \in B^{\prime}\end{cases}$$ The function $\theta$ is well-defined because $A^{\prime} \cap B^{\prime}=\varnothing$. We have described disjoint union categorically (i.e., with diagrams).

In Category Theory, we often view objects, not in isolation, but together with morphisms relating them to other objects; for example, objects may arise as solutions to universal mapping problems.

线性代数作业代写linear algebra代考|Limits

We now discuss inverse limit, a construction generalizing products, pullbacks, kernels, equalizers, and intersections, and direct limit, which generalizes coproducts, pushouts, cokernels, coequalizers, and unions.

Definition. Given a partially ordered set $I$ and a category $\mathcal{C}$, an inverse system in $\mathcal{C}$ is an ordered pair $\left(\left(M_{i}\right){i \in I},\left(\psi{i}^{j}\right){j \succeq i}\right)$, abbreviated $\left{M{i}, \psi_{i}^{j}\right}$, where $\left(M_{i}\right){i \in I}$ is an indexed family of objects in $\mathcal{C}$ and $\left(\psi{i}^{j}: M_{j} \rightarrow M_{i}\right){j \succeq i}$ is an indexed family of morphisms for which $\psi{i}^{i}=1_{M_{i}}$ for all $i$, and such that the following diagram commutes whenever $k \geq j \succeq i$.

A partially ordered set $I$, when viewed as a category, has as its objects the elements of $I$ and as its morphisms exactly one morphism $\kappa_{j}^{i}: i \rightarrow j$ whenever $i \preceq j$. It is easy to see that inverse systems in $\mathcal{C}$ over $I$ are merely contravariant functors $M: I \rightarrow \mathcal{C}$; in our original notation, $M(i)=M_{i}$ and $M\left(\kappa_{j}^{i}\right)=\psi_{i}^{j}$

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