## 数学代写|几何变换代写transformation geometry代考|МАТН5210

statistics-lab™ 为您的留学生涯保驾护航 在代写几何变换transformation geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写几何变换transformation geometry代写方面经验极为丰富，各种代写几何变换transformation geometry相关的作业也就用不着说。

• 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|几何变换代写transformation geometry代考|The Matrix of a Linear Transformation

We end this chapter on a point of great importance: that every linear transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$ amounts to multiplication by a matrix A. In this case, we say that $\mathbf{A}$ represents $T$ :

Definition 5.1. A linear transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$ is represented by a matrix A when we can compute $T$ using multiplication by $\mathbf{A}$. In other words, A represents $T$ when we have
$$T(\mathbf{x})=\mathbf{A} \mathbf{x}$$
for all inputs $\mathbf{x} \in \mathbf{R}^m$.
As the course proceeds, we’ll learn how to answer almost any question about a linear transformation-like the basic mapping questions listed in Section $3.6$ above – by analyzing the matrix that represents it. We’ll begin acquiring tools for that kind of analysis in Chapter 2. First though, we want to show how to find the matrix that represents a given linear map.
We start with Observation 1.12, which shows how to expand any vector $\mathbf{x}:=\left(x_1, x_2, \ldots, x_n\right) \in \mathbf{R}^m$ as a linear combination of standard basis vectors in a simple way:
$$\left(x_1, x_2, \ldots, x_n\right)=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_m \mathbf{e}_m$$
If we expand a vector $\mathbf{x}$ this way, and then map it into $\mathbf{R}^n$ using a linear transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$, the linearity rules (Definition 4.1) yield \begin{aligned} T(\mathbf{x}) & =T\left(x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_m \mathbf{e}_m\right) \ & =T\left(x_1 \mathbf{e}_1\right)+T\left(x_2 \mathbf{e}_2\right)+\cdots+T\left(x_m \mathbf{e}_m\right) \ & =x_1 T\left(\mathbf{e}_1\right)+x_2 T\left(\mathbf{e}_2\right)+\cdots+x_m T\left(\mathbf{e}_m\right) \end{aligned}
This reveals a powerful fact:

## 数学代写|几何变换代写transformation geometry代考|The Linear System

We now begin to focus on answering the basic mapping questions for linear transformations; that is, for linear mappings
$$T: \mathbf{R}^m \rightarrow \mathbf{R}^n$$
As we observed in Theorem 5.6, every linear transformation is represented by a matrix, via matrix/vector multiplication. Specifically, we have the formula
$$T(\mathbf{x})=\mathbf{A} \mathbf{x}$$
where $\mathbf{A}$ is the matrix whose columns are given by the $T\left(\mathbf{e}_j\right)$ ‘s. For this reason, we can usually reduce questions about the mapping $T$ to calculations involving the matrix $\mathbf{A}$.
In this chapter, we focus on the question of pre-image:
Problem: Given a linear transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$, and a point $\mathbf{b}$ in the range of $T$ how can we find all points in the pre-image $T^{-1}(\mathbf{b})$.

Since every linear map amounts to multiplication by a matrix (and conversely, multiplication by any matrix A defines a linear map), finding $T^{-1}(\mathbf{b})$ is the same as solving $T(\mathbf{x})=\mathbf{b}$ for $\mathbf{x}$. When $T$ is represented by $\mathbf{A}$, we have $T(\mathbf{x})=\mathbf{A} \mathbf{x}$, so the Problem above is exactly the same as this equivalent problem: Given an $n \times m$ matrix $\mathbf{A}$, and a vector $\mathbf{b} \in \mathbf{R}^n$, how can we find every $\mathbf{x} \in \mathbf{R}^m$ that solves the matrix/vector equation
$$\mathbf{A} \mathbf{x}=\mathbf{b}$$
This statement of the problem is nice and terse, but to solve it, we first need to expand its symbols in terms of matrix entries and coordinates. We start with $\mathbf{A}$.

As an $n \times m$ matrix, A has $n$ rows and $m$ columns. Doublesubscripting its entries in the usual way,

# 几何变换代考

## 数学代写|几何变换代写transformation geometry代考|The Matrix of a Linear Transformation

$$T(\mathbf{x})=\mathbf{A x}$$

$$\left(x_1, x_2, \ldots, x_n\right)=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_m \mathbf{e}_m$$

$$T(\mathbf{x})=T\left(x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_m \mathbf{e}_m\right) \quad=T\left(x_1 \mathbf{e}_1\right)+T\left(x_2 \mathbf{e}_2\right)+\cdots+T\left(x_m\right.$$

## 数学代写|几何变换代写transformation geometry代考|The Linear System

$$T: \mathbf{R}^m \rightarrow \mathbf{R}^n$$

$$T(\mathbf{x})=\mathbf{A} \mathbf{x}$$

$$\mathbf{A x}=\mathbf{b}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|几何变换代写transformation geometry代考|MATH319

statistics-lab™ 为您的留学生涯保驾护航 在代写几何变换transformation geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写几何变换transformation geometry代写方面经验极为丰富，各种代写几何变换transformation geometry相关的作业也就用不着说。

• 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|几何变换代写transformation geometry代考|Mappings and Transformations

The functions we study in Linear Algebra usually have domains and/or ranges in one of the numeric vector spaces $\mathbf{R}^n$ we introduced in Section 1.

DEFINITION 3.1. A function with numeric vector inputs or outputs is called a mapping or transformation – synonymous terms. A mapping, or transformation is thus simply a function described by a diagram of the form
$$F: \mathbf{R}^n \rightarrow \mathbf{R}^m$$
where $n>1$ and/or $m>1$. Typically, we use uppercase letters like $F$, $G$, or $H$ to label mappings, and from now on, we try to reserve the word function for the case of scalar outputs $(m=1)$.
Example 3.2. A simple mapping
$$J: \mathbf{R}^2 \rightarrow \mathbf{R}^2$$
is given by the rule
$$J(x, y)=(-y, x)$$
This formula makes it easy to compute $J(x, y)$ for any specific input $(x, y) \in \mathbf{R}^2$. For instance, we have
$$J(1,2)=(-2,1), \quad J(-3,5)=(-5,-3), \quad \text { and } \quad J(0,0)=(0,0)$$
Is $J$ one-to-one and/or onto? We leave that as part of Exercise 32 below.

While the domain and range of $J$ are the same, other mappings often have domains and ranges that differ, as the following examples illustrate.
Example $3.3$. The rule
$$F(x, y, z, w)=(x-y, z+w)$$
has four scalar entries in its input, but only two in its output.

## 数学代写|几何变换代写transformation geometry代考|Linearity

Recall that both scalar multiplication and matrix/vector multiplication distribute over vector addition (Propositions $1.6$ and 1.25). The definition of linearity generalizes those distributivity rules:

Definition 4.1. A mapping $F: \mathbf{R}^n \rightarrow \mathbf{R}^m$ is linear if it has both these properties:
i) F commutes with vector addition, meaning that for any two inputs $\mathbf{x}, \mathbf{y} \in \mathbf{R}^m$, we have
$$F(\mathbf{x}+\mathbf{y})=F(\mathbf{x})+F(\mathbf{y})$$
ii) $F$ commutes with scalar multiplication, meaning that for any input $\mathbf{x} \in \mathbf{R}^m$ and any scalar $c \in \mathbf{R}$, we have
$$F(c \mathbf{x})=c F(\mathbf{x})$$
Linear mappings are often called linear transformations, and for this reason the favorite symbol for a linear mapping is the letter $T$.
ExAmple $4.2$. The mapping $T: \mathbf{R}^2 \rightarrow \mathbf{R}^2$ given by
(2) $T(a, b)=(2 b, 3 a)$
is linear.
To verify this, we have to show that $T$ has both properties in Definition $4.1$ above.

First property: $T$ commutes with addition: We have to show that for any two vectors $\mathbf{x}=\left(x_1, x_2\right)$, and $\mathbf{y}=\left(y_1, y_2\right)$, we have
(3) $\quad T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$
We do so by expanding each side of the equation separately in coordinates, and checking that they give the same result. On the left, we have
$$T(\mathbf{x}+\mathbf{y})=T\left(\left(\begin{array}{l} x_1 \ x_2 \end{array}\right)+\left(\begin{array}{l} y_1 \ y_2 \end{array}\right)\right)=T\left(x_1+y_1, x_2+y_2\right)$$
and now the rule for $T$, namely (2), reduces this to $T(\mathbf{x}+\mathbf{y})=\left(2\left(x_2+y_2\right), 3\left(x_1+y_1\right)\right)=\left(2 x_2+2 y_2, 3 x_1+3 y_1\right)$

# 几何变换代考

## 数学代写|几何变换代写transformation geometry代考|Mappings and Transformations

$$F: \mathbf{R}^n \rightarrow \mathbf{R}^m$$

$$J: \mathbf{R}^2 \rightarrow \mathbf{R}^2$$

$$J(x, y)=(-y, x)$$

$$J(1,2)=(-2,1), \quad J(-3,5)=(-5,-3), \quad \text { and } \quad J(0,0)=(0,0)$$

$$F(x, y, z, w)=(x-y, z+w)$$

## 数学代写|几何变换代写transformation geometry代考|Linearity

i) $F$ 通过矢量加法交换，这意味着对于任何两个输入 $\mathbf{x}, \mathbf{y} \in \mathbf{R}^m$ ，我们有
$$F(\mathbf{x}+\mathbf{y})=F(\mathbf{x})+F(\mathbf{y})$$

$$F(c \mathbf{x})=c F(\mathbf{x})$$

(2)给出 $T(a, b)=(2 b, 3 a)$

(3) $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$

$$T(\mathbf{x}+\mathbf{y})=T\left(\left(x_1 x_2\right)+\left(y_1 y_2\right)\right)=T\left(x_1+y_1, x_2+y_2\right)$$

$$T(\mathbf{x}+\mathbf{y})=\left(2\left(x_2+y_2\right), 3\left(x_1+y_1\right)\right)=\left(2 x_2+2 y_2, 3 x_1+3 y_1\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|几何变换代写transformation geometry代考|MATH312

statistics-lab™ 为您的留学生涯保驾护航 在代写几何变换transformation geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写几何变换transformation geometry代写方面经验极为丰富，各种代写几何变换transformation geometry相关的作业也就用不着说。

• 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|几何变换代写transformation geometry代考|Numeric Vectors

The overarching goal of this book is to impart a sure grasp of the numeric vector functions known as linear transformations. Students will have encountered functions before. We review and expand that familiarity in Section 2 below, and we define linearity in Section 4. Before we can properly discuss these matters though, we must introduce numeric vectors and their basic arithmetic.

DEfinition $1.1$ (Vectors and scalars). A numeric vector (or just vector for short) is an ordered $n$-tuple of the form $\left(x_1, x_2, \ldots, x_n\right)$. Here, each $x_i$-the $i$ th entry (or $i$ th coordinate) of the vector-is a real number.

The $(x, y)$ pairs often used to label points in the plane are familiar examples of vectors with $n=2$, but we allow more than two entries as well. For instance, the triple $(3,-1 / 2,2)$, and the 7-tuple $(1,0,2,0,-2,0,-1)$ are also numeric vectors.
In the linear algebraic setting, we usually call single numbers scalars. This helps highlight the difference between numeric vectors and individual numbers.

Vectors can have many entries, so to clarify and save space, we often label them with single bold letters instead of writing out all their entries. For example, we might define
\begin{aligned} \mathbf{x} & :=\left(x_1, x_2, \ldots, x_n\right) \ \mathbf{a} & :=\left(a_1, a_2, a_3, a_4\right) \ \mathbf{b} & :=(-5,0,1) \end{aligned}
and then use $\mathbf{x}$, a, or $\mathbf{b}$ to indicate the associated vector. We use boldface to distinguish vectors from scalars. For instance, the same letters, without boldface, would typically represent scalars, as in $x=5$, $a=-4.2$, or $b=\pi$.
Often, we write numeric vectors vertically instead of horizontally, in which case $\mathbf{x}, \mathbf{a}$, and $\mathbf{b}$ above would look like this:

$$\mathbf{x}=\left(\begin{array}{r} x_1 \ x_2 \ \vdots \ x_m \end{array}\right), \quad \mathbf{a}=\left(\begin{array}{c} a_1 \ a_2 \ a_3 \ a_4 \end{array}\right), \quad \mathbf{b}=\left(\begin{array}{r} -5 \ 0 \ 1 \end{array}\right)$$
In our approach to the subject (unlike some others) we draw absolutely no distinction between
$$\left(x_1, x_2, \ldots, x_n\right) \text { and }\left(\begin{array}{r} x_1 \ x_2 \ \vdots \ x_n \end{array}\right)$$
These are merely different notations for the same vector – the very same mathematical object.

## 数学代写|几何变换代写transformation geometry代考|Functions

Now that we’re familiar with numeric vectors and matrices, we can consider vector functions – functions that take numeric vectors as inputs and produce them as outputs. The ultimate goal of this book is to give students a detailed understanding of linear vector functions, both algebraically, and geometrically. Here and in Section 3, we lay out the basic vocabulary for the kinds of questions one seeks to answer for any vector function, linear or not. Then, in Section 4, we introduce linearity, and with these building blocks all in place, we can at least state the main questions we’ll be answering in later chapters.
2.1. Domain, image, and range. Roughly speaking, a function is an input-output rule. Here is is a more precise formal definition.
DEFINITION 2.2. A function is an input/output relation specified by three data:
i) A domain set $X$ containing all allowed inputs,
ii) A range set $Y$ containing all allowed outputs, and
iii) A rule $f$ that assigns exactly one output $f(x)$ to every input $x$ in the domain.

We typically signal all three of these at once with a simple diagram like this:
$$f: X \rightarrow Y$$
For instance, if we apply the rule $T(x, y)=x+y$ to any input pair $(x, y) \in \mathbf{R}^2$, we get a scalar output in $\mathbf{R}$, and we can summarize this situation by writing $T: \mathbf{R}^2 \rightarrow \mathbf{R}$.

Technically, function and mapping are synonyms, but we will soon reserve the term function for the situation where (as with $T$ above) the range is just $\mathbf{R}$. When the range is $\mathbf{R}^n$ for some $n>1$, we typically prefer the term mapping or transformation.

# 几何变换代考

## 数学代写|几何变换代写transformation geometry代考|Numeric Vectors

$$\mathbf{x}:=\left(x_1, x_2, \ldots, x_n\right) \mathbf{a} \quad:=\left(a_1, a_2, a_3, a_4\right) \mathbf{b}:=(-5,0,1)$$

$$\mathbf{x}=\left(x_1 x_2 \vdots x_m\right), \quad \mathbf{a}=\left(a_1 a_2 a_3 a_4\right), \quad \mathbf{b}=\left(\begin{array}{lll} -5 & 0 & 1 \end{array}\right)$$

$$\left(x_1, x_2, \ldots, x_n\right) \text { and }\left(x_1 x_2 \vdots x_n\right)$$

## 数学代写|几何变换代写transformation geometry代考|Functions

2.1. 域 图像和范围。粗略地说，一个函数就是一个输入输出规则。这是一个更精确的正式定 义

i) 域集 $X$ 包含所有允许的输入，
ii) 范围集 $Y$ 包含所有允许的输出，以及
iii) 规则 $f$ 恰好分配一个输出 $f(x)$ 对每个输入 $x$ 在域中。

$$f: X \rightarrow Y$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。