# 线性代数网课代修|交换代数代写Commutative Algebra代考|MATH483

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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|Polynomials of One Variable

In this section, we will discuss polynomials of one variable and study the division algorithm from high school algebra. This simple algorithm has some surprisingly deep consequences-for example, we will use it to determine the structure of ideals of $k[x]$ and to explore the idea of a greatest common divisor. The theory developed will allow us to solve, in the special case of polynomials in $k[x]$, most of the problems raised in earlier sections. We will also begin to understand the important role played by algorithms.
By this point in their mathematics careers, most students have already seen a variety of algorithms, although the term “algorithm” may not have been used. Informally, an algorithm is a specific set of instructions for manipulating symbolic or numerical data. Examples are the differentiation formulas from calculus and the method of row reduction from linear algebra. An algorithm will have inputs, which are objects used by the algorithm, and outputs, which are the results of the algorithm. At each stage of execution, the algorithm must specify exactly what the next step will be.

When we are studying an algorithm, we will usually present it in “pseudocode,” which will make the formal structure easier to understand. Pseudocode is similar to the computer language Pascal, and a brief discussion is given in Appendix B. Another reason for using pseudocode is that it indicates how the algorithm could be programmed on a computer. We should also mention that most of the algorithms in this book are implemented in computer algebra systems such as AXIOM, Macsyma, Maple, Mathematica, and REDUCE. Appendix $\mathrm{C}$ has more details concerning these programs.

We begin by discussing the division algorithm for polynomials in $k[x]$. A crucial component of this algorithm is the notion of the “leading term” of a polynomial in one variable. The precise definition is as follows.

## 线性代数作业代写linear algebra代考|Orderings on the Monomials in k

If we examine in detail the division algorithm in $k[x]$ and the row-reduction (Gaussian elimination) algorithm for systems of linear equations (or matrices), we see that a notion of ordering of terms in polynomials is a key ingredient of both (though this is not often stressed). For example, in dividing $f(x)=x^{5}-3 x^{2}+1$ by $g(x)=x^{2}-4 x+7$ by the standard method, we would:

• Write the terms in the polynomials in decreasing order by degree in $x$.
• At the first step, the leading term (the term of highest degree) in $f$ is $x^{5}=x^{3} \cdot x^{2}=$ $x^{3} \cdot$ (leading term in $g$ ). Thus, we would subtract $x^{3} \cdot g(x)$ from $f$ to cancel the leading term, leaving $4 x^{4}-7 x^{3}-3 x^{2}+1$.
• Then, we would repeat the same process on $f(x)-x^{3} \cdot g(x)$, etc., until we obtain a polynomial of degree less than 2 .

For the division algorithm on polynomials in one variable, then we are dealing with the degree ordering on the one-variable monomials:
$$\cdots>x^{m+1}>x^{m}>\cdots>x^{2}>x>1 .$$
The success of the algorithm depends on working systematically with the leading terms in $f$ and $g$, and not removing terms “at random” from $f$ using arbitrary terms from $g$.
Similarly, in the row-reduction algorithm on matrices, in any given row, we systematically work with entries to the left first-leading entries are those nonzero entries farthest to the left on the row. On the level of linear equations, this is expressed by ordering the variables $x_{1}, \ldots, x_{n}$ as follows:
$$x_{1}>x_{2}>\cdots>x_{n} .$$

## 线性代数作业代写linear algebra代考|Orderings on the Monomials in k

• 将多项式中的项按度数降序写成 $x$.
• 第一步，在 $f$ 是 $x^{5}=x^{3} \cdot x^{2}=x^{3}$. (主导词在 $g$ )。因此，我们将减去 $x^{3} \cdot g(x)$ 从 $f$ 取 消前导词，离开 $4 x^{4}-7 x^{3}-3 x^{2}+1$.
• 然后，我们将重复相同的过程 $f(x)-x^{3} \cdot g(x)$ 等，直到我们得到一个次数小于 2 的多 项式。
对于单变量多项式的除法算法，我们处理单变量单项式的度数排序:
$$\cdots>x^{m+1}>x^{m}>\cdots>x^{2}>x>1 .$$
算法的成功取决于系统地使用 $f$ 和 $g$ ，而不是从“随机”中删除术语 $f$ 使用任意项 $g$.
类似地，在矩阵的行缩减算法中，在任何给定的行中，我们系统地使用左侧的条目，第一个 前导条目是该行最左侧的那些非零条目。在线性方程的水平上，这通过对变量进行排序来表 示 $x_{1}, \ldots, x_{n}$ 如下:
$$x_{1}>x_{2}>\cdots>x_{n}$$

# 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