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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|代数学代写Algebra代考|Linear Transformations as Matrices

One of the easiest ways to see that a function $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is indeed a linear transformation is to find a matrix whose multiplication has the same effect as T. For example,
if $A=\left[\begin{array}{cc}1 & -1 \ 1 & 1\end{array}\right]$ then $A \mathbf{v}=\left[\begin{array}{cc}1 & -1 \ 1 & 1\end{array}\right]\left[\begin{array}{l}v_1 \ v_2\end{array}\right]=\left[\begin{array}{l}v_1-v_2 \ v_1+v_2\end{array}\right]$,

which shows that multiplying a column vector by the matrix $A$ has the same effect as the linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined by $T\left(v_1, v_2\right)=\left(v_1-\right.$ $v_2, v_1+v_2$ ) (i.e., the linear transformation from Examples 1.4.1(c) and 1.4.2(b)). One of the most remarkable facts about linear transformations is that this procedure can always be carried out-every linear transformation can be represented via matrix multiplication, and there is a straightforward method for constructing a matrix that does the job:

A function $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a linear transformation if and only if there exists a matrix $[T] \in \mathcal{M}{m, n}$ such that $$T(\mathbf{v})=[T] \mathbf{v} \text { for all } \mathbf{v} \in \mathbb{R}^n .$$ Furthermore, the unique matrix $[T]$ with this property is called the standard matrix of $T$, and it is $$[T] \stackrel{\stackrel{d}{=}}{=}\left[T\left(\mathbf{e}_1\right)\left|T\left(\mathbf{e}_2\right)\right| \cdots \mid T\left(\mathbf{e}_n\right)\right] .$$ Proof. It follows immediately from Theorem 1.3.2 that if $[T] \in \mathcal{M}{m, n}$ then the function that sends $\mathbf{v}$ to $[T] \mathbf{v}$ is a linear transformation. We thus only have to prove that for every linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$, the matrix
$$[T]=\left[T\left(\mathbf{e}_1\right)\left|T\left(\mathbf{e}_2\right)\right| \cdots \mid T\left(\mathbf{e}_n\right)\right]$$
satisfies $[T] \mathbf{v}=T(\mathbf{v})$, and no other matrix has this property.

数学代写|代数学代写Algebra代考|A Catalog of Linear Transformations

In order to get more comfortable with the relationship between linear transformations and matrices, we construct the standard matrices of a few very geometrically-motivated linear transformations that come up frequently.
The two simplest linear transformations that exist are the zero transformation $O: \mathbb{R}^n \rightarrow \mathbb{R}^m$, defined by $O(\mathbf{v})=\mathbf{0}$ for all $\mathbf{v} \in \mathbb{R}^n$, and the identity transformation $I: \mathbb{R}^n \rightarrow \mathbb{R}^n$, defined by $I(\mathbf{v})=\mathbf{v}$ for all $\mathbf{v} \in \mathbb{R}^n$. It is perhaps not surprising that the standard matrices of these transformations are the zero matrix and the identity matrix, respectively. To verify this claim, just notice that if $O \in \mathcal{M}_{m, n}$ and $I \in \mathcal{M}_n$ are the zero matrix and the identity matrix, then $O \mathbf{v}=\mathbf{0}$ and $I \mathbf{v}=\mathbf{v}$ for all $\mathbf{v} \in \mathbb{R}^n$ too.
Diagonal Matrices
The next simplest type of linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is one that does not change the direction of the standard basis vectors, but just stretches them by certain (possibly different) amounts, as in Figure 1.15. These linear transformations are the ones for which there exist scalars $c_1, c_2, \ldots, c_n \in \mathbb{R}^n$ such that $T\left(v_1, v_2, \ldots, v_n\right)=\left(c_1 v_1, c_2 v_2, \ldots, c_n v_n\right)$.

代数学代写

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有限元方法代写

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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。

assignmentutor™您的专属作业导师
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