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# 数学代写|抽象代数代写Abstract Algebra代考|Math417 Matrix Groups

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## 数学代写|抽象代数代写Abstract Algebra代考|Groups ℤ10 and U10

Consider $\mathbb{Z}{10}={0,1,2,3,4,5,6,7,8,9}$. When working with a particular group, we use only one operation. However, $\mathbb{Z}{10}$ admits both addition and multiplication; i.e., $\mathbb{Z}{10}$ is closed under both operations. In fact, $\mathbb{Z}{10}$ is an example of a structure called a ring, which we will study much later in the textbook.

We’ve seen that $\mathbb{Z}{10}$ is a group under addition. (See Example $8.3$ for a justification.) But is $\mathbb{Z}{10}$ also a group under multiplication? Let’s check the group properties:
(1) $\mathbb{Z}{10}$ is closed under multiplication. (2) Multiplication in $\mathbb{Z}{10}$ is associative.
(3) The multiplicative identity element is $1 \in \mathbb{Z}{10}$, as $1 \cdot a=a$ and $a \cdot 1=a$ for all $a \in \mathbb{Z}{10}$
(4) Not every $a \in \mathbb{Z}{10}$ has a multiplicative inverse $a^{-1} \in \mathbb{Z}{10}$ such that $a \cdot a^{-1}=1$ and $a^{-1} \cdot a=1$

The last group property about inverses fails for $\mathbb{Z}_{10}$ with multiplication, as shown in the example below.

## 数学代写|抽象代数代写Abstract Algebra代考|Groups M(ℤ10 ) and G(ℤ10 )

Recall that $M\left(\mathbb{Z}{10}\right)$ is the set of $2 \times 2$ matrices with entries in $\mathbb{Z}{10}$. Moreover, $M\left(\mathbb{Z}{10}\right)$ admits both addition and multiplication; i.e., $M\left(\mathbb{Z}{10}\right)$ is closed under both operations. We’ll later see that $M\left(\mathbb{Z}_{10}\right)$ is also a ring.

Example 10.2. We’ll briefly review how to add and multiply in $M\left(\mathbb{Z}{10}\right)$. For more details, see Section 7.1. Let $\alpha, \beta \in M\left(\mathbb{Z}{10}\right)$, where $\alpha=\left[\begin{array}{cc}1 & 2 \ 3 & 4\end{array}\right]$ and $\beta=\left[\begin{array}{cc}5 & 6 \ 7 & 8\end{array}\right]$. Then
$$\alpha+\beta=\left[\begin{array}{ll} 1 & 2 \ 3 & 4 \end{array}\right]+\left[\begin{array}{ll} 5 & 6 \ 7 & 8 \end{array}\right]=\left[\begin{array}{ll} 1+5 & 2+6 \ 3+7 & 4+8 \end{array}\right]=\left[\begin{array}{ll} 6 & 8 \ 0 & 2 \end{array}\right]$$
and
$$\alpha \cdot \beta=\left[\begin{array}{ll} 1 & 2 \ 3 & 4 \end{array}\right] \cdot\left[\begin{array}{ll} 5 & 6 \ 7 & 8 \end{array}\right]=\left[\begin{array}{ll} 1 \cdot 5+2 \cdot 7 & 1 \cdot 6+2 \cdot 8 \ 3 \cdot 5+4 \cdot 7 & 3 \cdot 6+4 \cdot 8 \end{array}\right]=\left[\begin{array}{ll} 9 & 2 \ 3 & 0 \end{array}\right] .$$
Similar to $\mathbb{Z}{10}$, we’ve seen that $M\left(\mathbb{Z}{10}\right)$ is a group under addition. (See Section $7.2$ for a justification.) We now ask the same question that we posed earlier about $\mathbb{Z}_{10}$;

namely, is $M\left(\mathbb{Z}{10}\right)$ also a group under multiplication? We check the group properties: (1) $M\left(\mathbb{Z}{10}\right)$ is closed under multiplication.
(2) Multiplication in $M\left(\mathbb{Z}{10}\right)$ is associative. (See Chapter 】, Exercise #6.) (3) The multiplicative identity element is $\varepsilon=\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right]$, as $\varepsilon \cdot \alpha=\alpha$ and $\alpha \cdot \varepsilon=\alpha$ for all $\alpha \in M\left(\mathbb{Z}{10}\right)$
(4) Not every $\alpha \in M\left(\mathbb{Z}{10}\right)$ has a multiplicative inverse $\alpha^{-1} \in M\left(\mathbb{Z}{10}\right)$ such that $\alpha \cdot \alpha^{-1}=\varepsilon$ and $\alpha^{-1} \cdot \alpha=\varepsilon$

## 数学代写|抽象代数代写抽象代数代考|组ℤ10和U10

(1) $\mathbb{Z}{10}$ 对乘法封闭。(2)乘 $\mathbb{Z}{10}$
(3)乘法单位元为 $1 \in \mathbb{Z}{10}$，如 $1 \cdot a=a$ 和 $a \cdot 1=a$ 为所有人 $a \in \mathbb{Z}{10}$
(4)不是每一个 $a \in \mathbb{Z}{10}$ 有乘法逆吗 $a^{-1} \in \mathbb{Z}{10}$ 如此这般 $a \cdot a^{-1}=1$ 和 $a^{-1} \cdot a=1$

## 数学代写|抽象代数代写抽象代数代考|组M(ℤ10)和G(ℤ10)

$$\alpha+\beta=\left[\begin{array}{ll} 1 & 2 \ 3 & 4 \end{array}\right]+\left[\begin{array}{ll} 5 & 6 \ 7 & 8 \end{array}\right]=\left[\begin{array}{ll} 1+5 & 2+6 \ 3+7 & 4+8 \end{array}\right]=\left[\begin{array}{ll} 6 & 8 \ 0 & 2 \end{array}\right]$$

$$\alpha \cdot \beta=\left[\begin{array}{ll} 1 & 2 \ 3 & 4 \end{array}\right] \cdot\left[\begin{array}{ll} 5 & 6 \ 7 & 8 \end{array}\right]=\left[\begin{array}{ll} 1 \cdot 5+2 \cdot 7 & 1 \cdot 6+2 \cdot 8 \ 3 \cdot 5+4 \cdot 7 & 3 \cdot 6+4 \cdot 8 \end{array}\right]=\left[\begin{array}{ll} 9 & 2 \ 3 & 0 \end{array}\right] .$$

(2) $M\left(\mathbb{Z}{10}\right)$中的乘法是结合的。(见第6章)(3)乘的单位元是$\varepsilon=\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right]$，因为$\varepsilon \cdot \alpha=\alpha$和$\alpha \cdot \varepsilon=\alpha$对于所有$\alpha \in M\left(\mathbb{Z}{10}\right)$
(4)不是每个$\alpha \in M\left(\mathbb{Z}{10}\right)$都有一个乘的逆$\alpha^{-1} \in M\left(\mathbb{Z}{10}\right)$，这样$\alpha \cdot \alpha^{-1}=\varepsilon$和$\alpha^{-1} \cdot \alpha=\varepsilon$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。