# 数学代写|代数学代写Algebra代考|МATH611

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## 数学代写|代数学代写Algebra代考|Rotations in Higher Dimensions

While projections and reflections work regardless of the dimension of the space that they act on, extending rotations to dimensions higher than 2 is somewhat delicate. For example, in $\mathbb{R}^3$ it does not make sense to say something like “rotate $\mathbf{v}=(1,2,3)$ counter-clockwise by an angle of $\pi / 4 “$ “-this instruction is under-specified, since one angle alone does not tell us in which direction we should rotate.

In $\mathbb{R}^3$, we can get around this problem by specifying an axis of rotation via a unit vector (much like we used a unit vector to specify which line to project onto or reflect through earlier). However, we still have the problem that a rotation that looks clockwise from one side looks counter-clockwise from the other. Furthermore, there is a slightly simpler method that extends more straightforwardly to even higher dimensions-repeatedly rotate in a plane containing two of the coordinate axes.

To illustrate how to do this, consider the linear transformation $R_{y z}^\theta$ that rotates $\mathbb{R}^3$ by an angle of $\theta$ around the $x$-axis, from the positive $y$-axis toward the positive $z$-axis. It is straightforward to see that rotating $\mathbf{e}1$ in this way has no effect at all (i.e., $R{y z}^\theta\left(\mathbf{e}1\right)=\mathbf{e}_1$ ), as shown in Figure 1.20. Similarly, to rotate $\mathbf{e}_2$ or $\mathbf{e}_3$ we can just treat the $y z$-plane as if it were $\mathbb{R}^2$ and repeat the derivation from Figure $1.18$ to see that $R{y z}^\theta\left(\mathbf{e}2\right)=(0, \cos (\theta), \sin (\theta))$ and $R{y z}^\theta\left(\mathbf{e}_3\right)=(0,-\sin (\theta), \cos (\theta))$

## 数学代写|代数学代写Algebra代考|Composition of Linear Transformations

As we mentioned earlier, there are some simple ways of combining linear transformations to create new ones. In particular, we noted that if $S, T: \mathbb{R}^n \rightarrow$ $\mathbb{R}^m$ are linear transformations and $c \in \mathbb{R}$ then $S+c T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is also a linear transformation.

We now introduce another method of combining linear transformations that is slightly more exotic, and has the interpretation of applying one linear transformation after another:

Suppose $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $S: \mathbb{R}^m \rightarrow \mathbb{R}^p$ are linear transformations. Then their composition is the function $S \circ T: \mathbb{R}^n \rightarrow \mathbb{R}^p$ defined by
$$(S \circ T)(\mathbf{v})=S(T(\mathbf{v})) \quad \text { for all } \quad \mathbf{v} \in \mathbb{R}^n \text {. }$$
That is, the composition $S \circ T$ of two linear transformations is the function that has the same effect on vectors as first applying $T$ to them and then applying $S$. In other words, while $T$ sends $\mathbb{R}^n$ to $\mathbb{R}^m$ and $S$ sends $\mathbb{R}^m$ to $\mathbb{R}^p$, the composition $S \circ T$ skips the intermediate step and sends $\mathbb{R}^n$ directly to $\mathbb{R}^p$, as illustrated in Figure 1.21.

Even though $S$ and $T$ are linear transformations, at first it is not particularly obvious whether or not $S \circ T$ is also linear, and if so, what its standard matrix is. The following theorem shows that $S \circ T$ is indeed linear, and its standard matrix is simply the product of the standard matrices of $S$ and $T$. In fact, this theorem is the main reason that matrix multiplication was defined in the seemingly bizarre way that it was.

In other words, $S \circ T$ is a function that acts on $\mathbf{v}$ in the exact same way as matrix multiplication by the matrix $[S][T]$. It thus follows from Theorem 1.4.1 that $S \circ T$ is a linear transformation and its standard matrix is $[S][T]$, as claimed.
By applying linear transformations one after another like this, we can construct simple algebraic descriptions of complicated geometric transformations. For example, it might be difficult to visualize exactly what happens to $\mathbb{R}^2$ if we rotate it, reflect it, and then rotate it again, but this theorems tells us that we can unravel exactly what happens just by multiplying together the standard matrices of the three individual linear transformations.

# 代数学代写

## 数学代写|代数学代写Algebra代考| Higher Dimensions旋转

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$$(S \circ T)(\mathbf{v})=S(T(\mathbf{v})) \quad \text { for all } \quad \mathbf{v} \in \mathbb{R}^n \text {. }$$

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## 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。

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