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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|Row and Column Vectors

One of the most useful features of matrix multiplication is that it can be used to move vectors around. In particular, if $A \in \mathcal{M}{m, n}$ and $\mathbf{v} \in \mathcal{M}{n, 1}$ then $A \mathbf{v} \in \mathcal{M}_{m, 1}$, and since each of $\mathbf{v}$ and $A \mathbf{v}$ have just one column, we can think of them as vectors (and we think of $A$ as transforming $\mathbf{v}$ into $A \mathbf{v}$ ). In fact, we call these matrices with just one column column vectors, and similarly we refer to matrices with just one row as row vectors. For example,
if $A=\left[\begin{array}{ll}1 & 2 \ 3 & 4\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}2 \ -1\end{array}\right]$ then $A \mathbf{v}=\left[\begin{array}{l}2-2 \ 6-4\end{array}\right]=\left[\begin{array}{l}0 \ 2\end{array}\right]$, so we think of the matrix $A$ as transforming $\mathbf{v}=(2,-1)$ into $A \mathbf{v}=(0,2)$, but we have to write $\mathbf{v}$ and $A \mathbf{v}$ as columns for the matrix multiplication to actually work out.

When performing matrix multiplication (as well as a few other tasks that we will investigate later in this book), the difference between row vectors and column vectors is important, since the product $A \mathbf{v}$ does not make sense if $\mathbf{v}$ is a row vector-the inner dimensions of the matrix multiplication do not match. However, when thinking of vectors geometrically as we did in the previous sections, this difference is often unimportant.

To help make it clearer whether or not we actually care about the shape (i.e., row or column) of a vector, we use square brackets when thinking of a vector as a row or a column (as we have been doing throughout this section for matrices), and we use round parentheses if its shape is unimportant to us

(as we did in the previous two sections). For example, $\mathbf{v}=[1,2,3] \in \mathcal{M}{1,3}$ is a row vector, $$\mathbf{w}=\left[\begin{array}{l} 1 \ 2 \ 3 \end{array}\right] \in \mathcal{M}{3,1}$$
is a column vector, and $\mathbf{x}=(1,2,3) \in \mathbb{R}^3$ is a vector for which we do not care if it is a row or column. If the shape of a vector matters (e.g., if we multiply it by a matrix) and we give no indication otherwise, it is assumed to be a column vector from now on.

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

The final main ingredient of linear algebra, after vectors and matrices, are linear transformations: functions that act on vectors and that do not “mess up” vector addition and scalar multiplication. Despite being the most abstract and difficult object to grasp in this chapter, they are of paramount importance and permeate all of linear algebra, so the reader is encouraged to explore this section thoroughly.

A linear transformation is a function $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ that satisfies the following two properties:
a) $T(\mathbf{v}+\mathbf{w})=T(\mathbf{v})+T(\mathbf{w})$ for all vectors $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$, and
b) $T(c \mathbf{v})=c T(\mathbf{v})$ for all vectors $\mathbf{v} \in \mathbb{R}^n$ and all scalars $c \in \mathbb{R}$.
For example, it follows fairly quickly from Theorem 1.3.2 that matrix multiplication is a linear transformation-if $A \in \mathcal{M}_{m, n}$ then the function that sends $\mathbf{v} \in \mathbb{R}^n$ to $A \mathbf{v} \in \mathbb{R}^m$ preserves vector addition (i.e., $A(\mathbf{v}+\mathbf{w})=A \mathbf{v}+A \mathbf{w}$ ) and scalar multiplication (i.e., $A(c \mathbf{v})=c(A \mathbf{v})$ ).

Geometrically, linear transformations can be thought of as the functions that rotate, stretch, shrink, and/or reflect $\mathbb{R}^n$, but do so somewhat uniformly. For example, if we draw a square grid on $\mathbb{R}^2$ as in Figure 1.14, then a linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ can rotate, stretch, shrink, and/or reflect that grid, but it will still be made up of cells of the same size and shape (which in general will be parallelograms rather than squares). Furthermore, if a vector $\mathbf{v}$ is located in the $x$-th cell in the direction of $\mathbf{e}_1$ and the $y$-th cell in the direction of $\mathbf{e}_2$, then $T(\mathbf{v})$ is located in the $x$-th cell in the direction of $T\left(\mathbf{e}_1\right)$ and the $y$-th cell in the direction of $T\left(\mathbf{e}_2\right)$ (again, see Figure 1.14).

# 代数学代写

## 数学代写|代数学代写algebra代考|行和列向量

(正如我们在前两节中所做的那样)。例如，$\mathbf{v}=[1,2,3] \in \mathcal{M}{1,3}$是一个行向量，$$\mathbf{w}=\left[\begin{array}{l} 1 \ 2 \ 3 \end{array}\right] \in \mathcal{M}{3,1}$$

## 数学代写|代数学代写algebra代考|线性变换

. 线性代数的最后一个主要组成部分，在向量和矩阵之后，是线性变换:作用于向量的函数，不会“搞乱”向量加法和标量乘法。尽管它们是本章中最抽象和最难掌握的对象，但它们是最重要的，贯穿了所有线性代数，因此鼓励读者彻底探索本节 线性变换是一个函数 $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ 它满足以下两个属性:
a) $T(\mathbf{v}+\mathbf{w})=T(\mathbf{v})+T(\mathbf{w})$ 对于所有向量 $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$，和
b) $T(c \mathbf{v})=c T(\mathbf{v})$ 对于所有向量 $\mathbf{v} \in \mathbb{R}^n$ 所有的标量 $c \in \mathbb{R}$例如，根据定理1.3.2，矩阵乘法是一个线性变换——如果 $A \in \mathcal{M}_{m, n}$ 然后是发送的函数 $\mathbf{v} \in \mathbb{R}^n$ 到 $A \mathbf{v} \in \mathbb{R}^m$ 保留向量加法(即， $A(\mathbf{v}+\mathbf{w})=A \mathbf{v}+A \mathbf{w}$ )和标量乘法(即， $A(c \mathbf{v})=c(A \mathbf{v})$ ).

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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
assignmentutor™您的专属作业导师