# 线性代数网课代修|交换代数代写Commutative Algebra代考|MAT4200

linearalgebra.me 为您的留学生涯保驾护航 在线性代数linear algebra作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的线性代数linear algebra代写服务。我们的专家在线性代数linear algebra代写方面经验极为丰富，各种线性代数linear algebra相关的作业也就用不着 说。

• 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 and Affine Space

To link algebra and geometry, we will study polynomials over a field. We all know what polynomials are, but the term field may be unfamiliar. The basic intuition is that a field is a set where one can define addition, subtraction, multiplication, and division with the usual properties. Standard examples are the real numbers $\mathbb{R}$ and the complex numbers $\mathbb{C}$, whereas the integers $\mathbb{Z}$ are not a field since division fails ( 3 and 2 are integers, but their quotient $3 / 2$ is not). A formal definition of field may be found in Appendix A. One reason that fields are important is that linear algebra works over any field. Thus, even if your linear algebra course restricted the scalars to lie in $\mathbb{R}$ or $\mathbb{C}$, most of the theorems and techniques you learned apply to an arbitrary field $k$. In this book, we will employ different fields for different purposes. The most commonly used fields will be:

• The rational numbers $\mathbb{Q}$ : the field for most of our computer examples.
• The real numbers $\mathbb{R}$ : the field for drawing pictures of curves and surfaces.
• The complex numbers $\mathbb{C}$ : the field for proving many of our theorems. On occasion, we will encounter other fields, such as fields of rational functions (which will be defined later). There is also a very interesting theory of finite fields-see the exercises for one of the simpler examples.

We can now define polynomials. The reader certainly is familiar with polynomials in one and two variables, but we will need to discuss polynomials in $n$ variables $x_{1}, \ldots, x_{n}$ with coefficients in an arbitrary field $k$. We start by defining monomials.

## 线性代数作业代写linear algebra代考|Parametrizations of Affine Varieties

In this section, we will discuss the problem of describing the points of an affine variety $\mathbf{V}\left(f_{1}, \ldots, f_{s}\right)$. This reduces to asking whether there is a way to “write down” the solutions of the system of polynomial equations $f_{1}=\cdots=f_{s}=0$. When there are finitely many solutions, the goal is simply to list them all. But what do we do when there are infinitely many? As we will see, this question leads to the notion of parametrizing an affine variety.

To get started, let us look at an example from linear algebra. Let the field be $\mathbb{R}$, and consider the system of equations $$\begin{gathered} x+y+z=1 \ x+2 y-z=3 . \end{gathered}$$ Geometrically, this represents the line in $\mathbb{R}^{3}$ which is the intersection of the planes $x+y+z=1$ and $x+2 y-z=3$. It follows that there are infinitely many solutions. To describe the solutions, we use row operations on equations (1) to obtain the equivalent equations \begin{aligned} &x+3 z=-1 \ &y-2 z=2 \end{aligned} Letting $z=t$, where $t$ is arbitrary, this implies that all solutions of (1) are given by \begin{aligned} &x=-1-3 t, \ &y=2+2 t, \ &z=t \end{aligned} as $t$ varies over $\mathbb{R}$. We call $t$ a parameter, and (2) is, thus, a parametrization of the solutions of (1).

## 线性代数作业代写linear algebra代考|Polynomials and Affine Space

• 有理数问：我们大多数计算机示例的字段。
• 真实的数字R：绘制曲线和曲面图片的领域。
• 复数C: 证明我们的许多定理的领域。 有时，我们会遇到其他领域，例如有理函数领域（稍后将定义）。还有一个非常有趣的有限域理论——参见练习中的一个更简单的例子。

## 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