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# 数学代写|抽象代数代写Abstract Algebra代考|MATH413 Matrices and transformations

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## 数学代写|抽象代数代写Abstract Algebra代考|Matrices and transformations

Another type of object is a matrix. Matrices, like functions, might feel less like objects than numbers. But matrices of appropriately matching sizes can be combined via binary operations. For instance, for $2 \times 2$ matrices, addition and multiplication work like this:
\begin{aligned} \left(\begin{array}{ll} a & b \ c & d \end{array}\right)+\left(\begin{array}{ll} k & l \ m & n \end{array}\right) & =\left(\begin{array}{ll} a+k & b+l \ c+m & d+n \end{array}\right) \ \left(\begin{array}{ll} a & b \ c & d \end{array}\right)\left(\begin{array}{ll} k & l \ m & n \end{array}\right) & =\left(\begin{array}{ll} a k+b m & a l+b n \ c k+d m & c l+d m \end{array}\right) . \end{aligned}
The set of $2 \times 2$ matrices with entries in $\mathbb{R}$ is sometimes denoted by $M_2(\mathbb{R})$ or $M(2, \mathbb{R})$ or $M_{2 \times 2}(\mathbb{R})$ (I know, multiple notations are annoying-I prefer the last as it seems clearest what it means). This set is closed under matrix addition and under matrix multiplication-why? Both operations are associative, which you can check as in Section 1.2. Addition is commutative, but multiplication is not, so $M_{2 \times 2}(\mathbb{R})$ under multiplication is a non-commutative algebraic structure.

How about identities and inverses? For addition, the identity is a matrix of zeros, and all $2 \times 2$ matrices have additive inverses. For multiplication, the identity is
$$\left(\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right) \text {, and }\left(\begin{array}{ll} a & b \ c & d \end{array}\right) \text { has inverse } \frac{1}{a d-b c}\left(\begin{array}{cc} d & -b \ -c & a \end{array}\right) \text {. }$$

## 数学代写|抽象代数代写Abstract Algebra代考|Symmetries and permutations

Another type of mathematical object is a symmetry. The idea of symmetries as objects was introduced in Section 1.3, which noted that an equilateral triangle has six distinct symmetries: two rotations, three reflections and an identity (‘do nothing’). These symmetries are represented below; the dots are just to track which vertex goes where.

Each symmetry can be understood as a transformation, a function mapping the triangle to itself. We could think of the triangle as centred at $(0,0)$ so that its symmetries form a subset of isometries of the plane. Indeed, you might observe that the triangle in fact has infinitely many symmetries because we could keep spinning it: a rotation through $480^{\circ}$ would map it to itself just as well as a rotation through $120^{\circ}$. But the two have the same effect on the triangle’s vertices, so there are only six interestingly different symmetries.

Because symmetries are transformations, they can be combined using composition. For the symmetries of an equilateral triangle, we can construct a binary operation table by cutting out a triangle, labelling its vertices, deciding on a ‘start’ position, then performing pairs of symmetries to check where it ends up. For instance, as observed in Section 1.3, the rotation $\rho$ followed by the reflection $r_1$ gives the single reflection $r_2$.

I strongly recommend that you do this cutting, labelling and turning. I am not a stickler for tedious checks, but I do believe that developing intuition for a new structure requires more than just reading a book. It will take five minutes to construct the whole table, so please do. Then check that it matches that below.

## 数学代写|抽象代数代写Abstract Algebra代考|Matrices and transformations

$\left(\begin{array}{llll}1 & 0 & 0 & 1\end{array}\right)$, and $\left(\begin{array}{llll}a & b c & d\end{array}\right)$ has inverse $\frac{1}{a d-b c}\left(\begin{array}{lll}d & -b-c & a\end{array}\right)$.

## 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。