## 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。

## 数学代写|数学分析代写Mathematical Analysis代考|MATH2050

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical Analysis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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis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|数学分析代写Mathematical Analysis代考|Power Series

Let $a_k, k=0,1,2, \ldots$, be a sequence of real numbers. The series of functions
$$a_0+a_1 x+a_2 x^2+\ldots+a_k x^k+\ldots$$
is called power series with coefficients $a_0, a_1, a_2, \ldots, a_k, \ldots$
A power series satisfies one of the following:
(i) the series converges only at $x=0$;
(ii) the series converges at any $x \in \mathbb{R}$;
(iii) there exists a real number $\varrho>0$ such that the series converges for $|x|<\varrho$ and does not converge for $|x|>\varrho$.

In particular, the convergence set of the power series (1.24), i.e. the set of points $x \in \mathbb{R}$ at which (1.24) converges, is an interval centred at the origin, namely: just ${0}$ in case (i), the whole $\mathbb{R}$ in case (ii), and an interval between $-\varrho, \varrho$ in case (iii). To prove these claims let us begin with the following result.

Theorem 1 If the power series (1.24) converges at some $\xi \neq 0$, it converges totally on any closed, bounded interval contained in $(-|\xi|,|\xi|)$.
Proof The convergence of the numerical series
$$a_0+a_1 \xi+a_2 \xi^2+\ldots+a_k \xi^k+\ldots$$
implies that the sequence $a_k \xi^k$ is infinitesimal as $k \rightarrow+\infty$, and hence bounded. Put equivalently, there exists $M>0$ such that
$$\left|a_k \xi^k\right| \leq M, \quad \forall k \in \mathbb{N} .$$

## 数学代写|数学分析代写Mathematical Analysis代考|Taylor Series

Let $f(x)$ be a real function defined on an interval $(a, b)$ in $\mathbb{R}$ and let $x_0 \in(a, b)$ be a point. We seek to establish whether there exists a power series centred at $x_0$ that converges on $(a, b)$ to $f$, which is usually phrased by saying that $f$ can be expanded in power series around $x_0$ on the interval $(a, b)$.
The first result in this direction goes as follows.
Theorem 1 If the power series
$$\sum_{k=0}^{\infty} a_k\left(x-x_0\right)^k$$
has convergence radius $\varrho>0$, its sum $f(x)$ is differentiable infinitely many times for $\left|x-x_0\right|<Q$. and for any $m \in \mathbb{N}$ the mth derivative equals
$$f^{(m)}(x)=\sum_{k=m}^{\infty} k(k-1) \cdots(k-m+1) a_k\left(x-x_0\right)^{k-m}$$
Furthermore, $f$ admits a series expansion of the form
$$f(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}\left(x_0\right)}{k !}\left(x-x_0\right)^k$$
Proof Formula (1.36) arises by repeatedly applying Theorem 6 of the previous section (in particular, the theorem on differentiating power series). Putting $x=x_0$ in (1.36), all terms after the first vanish, and $f^{(m)}\left(x_0\right)=m ! a_m$ for every $m \in \mathbb{N}$. Substituting $a_k=f^{(k)}\left(x_0\right) / k$ ! in (1.35) gives (1.37).

By Theorem 1 we know that if $f$ can be expanded in power series around $x_0$ on $(a, b)$, then on some neighbourhood of $x_0$ inside $(a, b)$, of the form $\left|x-x_0\right|<\varrho$, we necessarily have that
(i) $f$ is differentiable infinitely many times when $\left|x-x_0\right|<\varrho$;

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Power Series

$$a_0+a_1 x+a_2 x^2+\ldots+a_k x^k+\ldots$$

(i) 该级数仅收敛于 $x=0$;
(ii) 该系列收敛于任何 $x \in \mathbb{R}$;
(iii) 存在实数 $\varrho>0$ 使得该系列收敛于 $|x|<\varrho$ 并且不收敛于 $|x|>\varrho$.

$$a_0+a_1 \xi+a_2 \xi^2+\ldots+a_k \xi^k+\ldots$$

$$\left|a_k \xi^k\right| \leq M, \quad \forall k \in \mathbb{N}$$

## 数学代写|数学分析代写Mathematical Analysis代考|Taylor Series

$$\sum_{k=0}^{\infty} a_k\left(x-x_0\right)^k$$

$$f^{(m)}(x)=\sum_{k=m}^{\infty} k(k-1) \cdots(k-m+1) a_k\left(x-x_0\right)^{k-m}$$

$$f(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}\left(x_0\right)}{k !}\left(x-x_0\right)^k$$

(i) $f$ 可微分无数次当 $\left|x-x_0\right|<\varrho$;

## 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|数学分析代写Mathematical Analysis代考|MATH2060

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical Analysis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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis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|数学分析代写Mathematical Analysis代考|Uniform Convergence and Monotonicity

In this section we shall discuss two classical results regarding uniform convergence under a monotonicity hypothesis. The first theorem (by Dini) assumes monotonicity in the parameter $k$, the second one supposes monotonicity in the variable $x$.

Theorem 1 (Dini) Let $I=[a, b]$ be a closed and bounded interval and consider a sequence $f_k: I \rightarrow \mathbb{R}$ of continuous functions, monotone in $k$ (for instance, increasing: $f_k(x) \leq f_{k+1}(x)$ for any $\left.k \in \mathbb{N}, x \in I\right)$, and pointwise convergent on $[a, b]$ to some continuous function $f$. Then $f_k$ converges uniformly to $f$ on $[a, b]$.
Proof Consider, for example, an increasing sequence $f_k$, that is to say $f_k(x) \leq$ $f_{k+1}(x) \leq f(x)$ for any $k \in \mathbb{N}$ and any $x \in I=[a, b]$.

Suppose, by contradiction, that $f_k$ does not converge uniformly to $f$ on $[a, b]$. This means there exists $\varepsilon_0>0$ such that for any $v \in \mathbb{N}$ we can find $k>v$ and $x \in[a, b]$ for which
$$\left|f_k(x)-f(x)\right|=f(x)-f_k(x) \geq \varepsilon_0 .$$
Hence for any $v=h \in \mathbb{N}$, there exist $k_h \rightarrow+\infty$ and $x_h \in[a, b]$ such that
$$f\left(x_h\right)-f_{k_h}\left(x_h\right) \geq \varepsilon_0 .$$
But the monotonicity of $f_k$ in $k$ forces $f_{k_h} \geq f_i$ when $k_h \geq i$. So we obtain
$$f\left(x_h\right)-f_i\left(x_h\right) \geq \varepsilon_0, \quad \forall h \in \mathbb{N}, \quad \forall i \leq k_h .$$
The sequence $x_h$, being bounded, admits a subsequence $x_{h_j}$ converging to a point $x_0$ of the interval $[a, b]$. Taking the limit as $j \rightarrow+\infty$ in
$$f\left(x_{h_j}\right)-f_i\left(x_{h_j}\right) \geq \varepsilon_0, \quad \forall h_j \in \mathbb{N}, \quad \forall i \leq k_{h_j},$$
due to the continuity of $f$ and $f_i$ we have
$$f\left(x_0\right)-f_i\left(x_0\right) \geq \varepsilon_0 \quad \forall i \in \mathbb{N} .$$
Taking the limit when $i \rightarrow+\infty$ we reach the contradiction $0 \geq \varepsilon_0$.

## 数学代写|数学分析代写Mathematical Analysis代考|Series of Functions

If $f_k$ is a sequence of real functions defined on the subset $I$ of $\mathbb{R}$, we indicate by $s_k$ the sequence of partial sums
\begin{aligned} & s_1=f_1 \ & s_2=f_1+f_2 \ & \ldots \ldots \ldots \ & s_k=f_1+f_2+\ldots+f_k \ & \ldots \ldots \ldots \ldots \end{aligned}
The sequence of functions $s_k$ is called series (of functions) with general term $f_k$, and we shall also use for it the expression
$$f_1+f_2+\ldots+f_k+\ldots$$
If, for any $x \in I$, the numerical series with general term $f_k(x)$
$$f_1(x)+f_2(x)+\ldots+f_k(x)+\ldots$$
is convergent, i.e. if the sequence $s_k(x)$ converges (it has finite limit) for every $x \in I$, one says that the series of functions (1.18) converges pointwise on $I$.

When the sequence of functions $s_k$ converges uniformly on $I$, we say the series of functions (1.18) converges uniformly on $I$. In either case, the limit of $s_k$ as $k \rightarrow+\infty$ is called sum of the series of general term $f_k$, and we denote it by
$$\sum_{k=1}^{\infty} f_k$$
At times, (1.19) also indicates the series of general term $f_k$, apart from its sum. Often, as in the case of numerical series, one uses distinct summation indices for a series’ general term and (for example) the sequence of partial sums of a convergent series:
\begin{aligned} s_k & =\sum_{i=1}^k f_i, \quad \forall k \in \mathbb{N} \ f & =\lim {k \rightarrow+\infty} s_k=\lim {k \rightarrow+\infty} \sum_{i=1}^k f_i=\sum_{i=1}^{\infty} f_i \end{aligned}

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Uniform Convergence and Monotonicity

$$\left|f_k(x)-f(x)\right|=f(x)-f_k(x) \geq \varepsilon_0 .$$

$$f\left(x_h\right)-f_{k_h}\left(x_h\right) \geq \varepsilon_0 .$$

$$f\left(x_h\right)-f_i\left(x_h\right) \geq \varepsilon_0, \quad \forall h \in \mathbb{N}, \quad \forall i \leq k_h .$$

$$f\left(x_{h_j}\right)-f_i\left(x_{h_j}\right) \geq \varepsilon_0, \quad \forall h_j \in \mathbb{N}, \quad \forall i \leq k_{h_j},$$

$$f\left(x_0\right)-f_i\left(x_0\right) \geq \varepsilon_0 \quad \forall i \in \mathbb{N} .$$

## 数学代写|数学分析代写Mathematical Analysis代考|Series of Functions

$$s_1=f_1 \quad s_2=f_1+f_2 \ldots \ldots . \quad s_k=f_1+f_2+\ldots+f_k \ldots \ldots \ldots \ldots$$

$$f_1+f_2+\ldots+f_k+\ldots$$

$$f_1(x)+f_2(x)+\ldots+f_k(x)+\ldots$$

$$\sum_{k=1}^{\infty} f_k$$

$$s_k=\sum_{i=1}^k f_i, \quad \forall k \in \mathbb{N} f \quad=\lim k \rightarrow+\infty s_k=\lim k \rightarrow+\infty \sum_{i=1}^k f_i=\sum_{i=1}^{\infty} f_i$$

## 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|数学分析代写Mathematical Analysis代考|MATH307

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical Analysis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。我们的专家在代写数学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis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|数学分析代写Mathematical Analysis代考|Sequences of Functions: Pointwise and Uniform Convergence

Let $I$ be a set of real numbers and $f_k: I \rightarrow \mathbb{R}$ a sequence of real functions defined on $I$. One says that $f_k$ converges to the function $f: I \rightarrow \mathbb{R}$ pointwise on $I$ whenever
$$\lim {k \rightarrow+\infty} f_k(x)=f(x), \quad \forall x \in I .$$ In other words, if for any $\varepsilon>0$ and any $x \in I$ there exists $v{\varepsilon, x} \in \mathbb{N}$ such that
$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v_{\varepsilon, x} .$$
In general, given $\varepsilon>0$, the number $v_{\varepsilon, x}$ depends on the point $x$; if, instead, this number is independent of $x$, one speaks of uniform convergence.

Precisely, we say that $f_k$ converges uniformly on $I$ to $f$ if, for any $\varepsilon>0$, there exists $v_{\varepsilon} \in \mathbb{N}$ such that
$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v_{\varepsilon}, \quad \forall x \in I .$$
Equivalently, $f_k$ converges uniformly on $I$ to $f$ if, for any $\varepsilon>0$, there exists $v_{\varepsilon} \in \mathbb{N}$ such that
$$\sup \left{\left|f_k(x)-f(x)\right|: x \in I\right}<\varepsilon, \quad \forall k>v_{\varepsilon} .$$
Another way to express the same is saying that $f_k$ converges uniformly on $I$ to $f$ if the following condition on the limit of a numerical sequence holds
$$\lim _{k \rightarrow+\infty} \sup \left{\left|f_k(x)-f(x)\right|: x \in I\right}=0 .$$

## 数学代写|数学分析代写Mathematical Analysis代考|First Theorems on Uniform Convergence

Let us start by describing the continuity property of the uniform limit of continuous functions. Suppose $f_k: I \subseteq \mathbb{R} \rightarrow \mathbb{R}$ is a sequence of continuous functions on the subset $I$ of $\mathbb{R}$, and assume that $f_k$ converges uniformly on $I$ to the function $f: I \rightarrow \mathbb{R}$; we shall prove that $f$ is continuous on $I$. Observe that this result does not hold if we only assume that $f_k$ converges to $f$ pointwise. This is what happens, for instance, in Example 2 of the previous section, where the discontinuous function (1.2) is the pointwise limit of the sequence of continuous functions (1.1).
Theorem (Continuity of Limits) Let $f_k: I \subseteq \mathbb{R} \rightarrow \mathbb{R}$ be a sequence of continuous functions that converges uniformly on I to the function $f$. Then $f$ is continuous.

Proof Let us verify that $f$ is continuous at $x_0$, for any given $x_0 \in I$. By the uniform convergence hypothesis, given $\varepsilon>0$, there exists $v$ such that
$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v, \quad \forall x \in I .$$
Let us choose $k_0>v$; then clearly for any $x \in I$ we have
\begin{aligned} \left|f(x)-f\left(x_0\right)\right| & \leq\left|f(x)-f_{k_0}(x)\right|+\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|+\left|f_{k_0}\left(x_0\right)-f\left(x_0\right)\right| \leq \ & <\varepsilon+\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|+\varepsilon . \end{aligned} Because of the continuity of $f_{k_0}$ it is possible to find $\delta>0$ such that
$$x \in I, \quad\left|x-x_0\right|<\delta \Rightarrow\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|<\varepsilon$$
and so for $x \in I,\left|x-x_0\right|<\delta$, we obtain
$$\left|f(x)-f\left(x_0\right)\right|<3 \varepsilon .$$
More generally, we have the following result.

## 数学代写|数学分析代写Mathematical Analysis代考|Sequences of Functions: Pointwise and Uniform Convergence

$$\lim k \rightarrow+\infty f_k(x)=f(x), \quad \forall x \in I .$$

$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v_{\varepsilon, x} .$$

$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v_{\varepsilon}, \quad \forall x \in I .$$

## 数学代写|数学分析代写Mathematical Analysis代考|First Theorems on Uniform Convergence

$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v, \quad \forall x \in I .$$

$$\left|f(x)-f\left(x_0\right)\right| \leq\left|f(x)-f_{k_0}(x)\right|+\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|+\left|f_{k_0}\left(x_0\right)-f\left(x_0\right)\right| \leq \quad<\varepsilon+\mid f_{k_0}$$ 因为连续性 $f_{k_0}$ 有可能找到 $\delta>0$ 这样
$$x \in I, \quad\left|x-x_0\right|<\delta \Rightarrow\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|<\varepsilon$$

$$\left|f(x)-f\left(x_0\right)\right|<3 \varepsilon .$$

## 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|线性代数代写linear algebra代考|MTH2106

statistics-lab™ 为您的留学生涯保驾护航 在代写线性代数linear algebra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。我们的专家在代写线性代数linear algebra代写方面经验极为丰富，各种代写线性代数linear algebra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|线性代数代写linear algebra代考|The Dimension of a Subspace

It can be shown that if a subspace $H$ has a basis of $p$ vectors, then every basis of $H$ must consist of exactly $p$ vectors. (See Exercises 27 and 28 .) Thus the following definition makes sense.
The dimension of a nonzero subspace $H$, denoted by $\operatorname{dim} H$, is the number of vectors in any basis for $H$. The dimension of the zero subspace ${0}$ is defined to be zero. ${ }^2$
The space $\mathbb{R}^n$ has dimension $n$. Every basis for $\mathbb{R}^n$ consists of $n$ vectors. A plane through 0 in $\mathbb{R}^3$ is two-dimensional, and a line through $\mathbf{0}$ is one-dimensional.

EXAMPLE 2 Recall that the null space of the matrix $A$ in Example 6 in Section $2.8$ had a basis of 3 vectors. So the dimension of $\operatorname{Nul} A$ in this case is 3 . Observe how each basis vector corresponds to a free variable in the equation $A \mathbf{x}=\mathbf{0}$. Our construction always produces a basis in this way. So, to find the dimension of $\mathrm{Nul} A$, simply identify and count the number of free variables in $A \mathbf{x}=\mathbf{0}$.
The rank of a matrix $A$, denoted by rank $A$, is the dimension of the column space of $A$.
Since the pivot columns of $A$ form a basis for $\operatorname{Col} A$, the rank of $A$ is just the number of pivot columns in $A$.

The row reduction in Example 3 reveals that there are two free variables in $A \mathbf{x}=\mathbf{0}$, because two of the five columns of $A$ are not pivot columns. (The nonpivot columns correspond to the free variables in $A \mathbf{x}=\mathbf{0}$.) Since the number of pivot columns plus the number of nonpivot columns is exactly the number of columns, the dimensions of Col $A$ and $\mathrm{Nul} A$ have the following useful connection. (See the Rank Theorem in Section $4.6$ for additional details.)
The Rank Theorem
If a matrix $A$ has $n$ columns, then $\operatorname{rank} A+\operatorname{dim} \operatorname{Nul} A=n$.
The following theorem is important for applications and will be needed in Chapters 5 and 6. The theorem (proved in Section 4.5) is certainly plausible, if you think of a $p$-dimensional subspace as isomorphic to $\mathbb{R}^p$. The Invertible Matrix Theorem shows that $p$ vectors in $\mathbb{R}^p$ are linearly independent if and only if they also span $\mathbb{R}^p$.

## 数学代写|线性代数代写linear algebra代考|Column Space and Null Space of a Matrix

Subspaces of $\mathbb{R}^n$ usually occur in applications and theory in one of two ways. In both cases, the subspace can be related to a matrix.
The column space of a matrix $A$ is the set $\operatorname{Col} A$ of all linear combinations of the columns of $A$.
If $A=\left[\begin{array}{lll}\mathbf{a}_1 & \cdots & \mathbf{a}_n\end{array}\right]$, with the columns in $\mathbb{R}^m$, then $\operatorname{Col} A$ is the same as Span $\left{\mathbf{a}_1, \ldots, \mathbf{a}_n\right}$. Example 4 shows that the column space of an $\boldsymbol{m} \times \boldsymbol{n}$ matrix is a subspace of $\mathbb{R}^m$. Note that $\operatorname{Col} A$ equals $\mathbb{R}^m$ only when the columns of $A$ span $\mathbb{R}^m$. Otherwise, $\operatorname{Col} A$ is only part of $\mathbb{R}^m$.

EXAMPLE 4 Let $A=\left[\begin{array}{rrr}1 & -3 & -4 \ -4 & 6 & -2 \ -3 & 7 & 6\end{array}\right]$ and $\mathbf{b}=\left[\begin{array}{r}3 \ 3 \ -4\end{array}\right]$. Determine whether $\mathbf{b}$ is in the column space of $A$.

SOLUTION The vector $\mathbf{b}$ is a linear combination of the columns of $A$ if and only if $\mathbf{b}$ can be written as $A \mathbf{x}$ for some $\mathbf{x}$, that is, if and only if the equation $A \mathbf{x}=\mathbf{b}$ has a solution. Row reducing the augmented matrix $\left[A \begin{array}{ll}A & \mathbf{b}\end{array}\right]$,
$$\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ -4 & 6 & -2 & 3 \ -3 & 7 & 6 & -4 \end{array}\right] \sim\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ 0 & -6 & -18 & 15 \ 0 & -2 & -6 & 5 \end{array}\right] \sim\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ 0 & -6 & -18 & 15 \ 0 & 0 & 0 & 0 \end{array}\right]$$
we conclude that $A \mathbf{x}=\mathbf{b}$ is consistent and $\mathbf{b}$ is in $\operatorname{Col} A$.

The solution of Example 4 shows that when a system of linear equations is written in the form $A \mathbf{x}=\mathbf{b}$, the column space of $A$ is the set of all $\mathbf{b}$ for which the system has a solution.
The null space of a matrix $A$ is the set $\mathrm{Nul} A$ of all solutions of the homogeneous equation $A \mathbf{x}=\mathbf{0}$
When $A$ has $n$ columns, the solutions of $A \mathbf{x}=\mathbf{0}$ belong to $\mathbb{R}^n$, and the null space of $A$ is a subset of $\mathbb{R}^n$. In fact, $\mathrm{Nul} A$ has the properties of a subspace of $\mathbb{R}^n$.
The null space of an $m \times n$ matrix $A$ is a subspace of $\mathbb{R}^n$. Equivalently, the set of all solutions of a system $A \mathbf{x}=\mathbf{0}$ of $m$ homogeneous linear equations in $n$ unknowns is a subspace of $\mathbb{R}^n$.
PROOF The zero vector is in $\operatorname{Nul} A$ (because $A 0=0$ ). To show that $\mathrm{Nul} A$ satisfies the other two properties required for a subspace, take any $\mathbf{u}$ and $\mathbf{v}$ in $\mathrm{Nul} A$. That is, suppose $A \mathbf{u}=\mathbf{0}$ and $A \mathbf{v}=\mathbf{0}$. Then, by a property of matrix multiplication,
$$A(\mathbf{u}+\mathbf{v})=A \mathbf{u}+A \mathbf{v}=\mathbf{0}+\mathbf{0}=\mathbf{0}$$
Thus $\mathbf{u}+\mathbf{v}$ satisfies $A \mathbf{x}=\mathbf{0}$, and so $\mathbf{u}+\mathbf{v}$ is in $\operatorname{Nul} A$. Also, for any scalar $c, A(c \mathbf{u})=$ $c(A \mathbf{u})=c(0)=\mathbf{0}$, which shows that $c \mathbf{u}$ is in $\mathrm{Nul} A$.

To test whether a given vector $\mathbf{v}$ is in $\operatorname{Nul} A$, just compute $A \mathbf{v}$ to see whether $A \mathbf{v}$ is the zero vector. Because $\mathrm{Nul} A$ is described by a condition that must be checked for each vector, we say that the null space is defined implicitly. In contrast, the column space is defined explicitly, because vectors in Col A can be constructed (by linear combinations) from the columns of $A$. To create an explicit description of $\mathrm{Nul} A$, solve the equation $A \mathbf{x}=\mathbf{0}$ and write the solution in parametric vector form. (See Example 6 , below.) ${ }^2$

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Column Space and Null Space of a Matrix

Veft{ $\left.\backslash m a t h b f{a} _1, \backslash d o t s, \backslash m a t h b f{a} _n \backslash r i g h t\right}$. 示例 4 显示了一个列空间 $\boldsymbol{m} \times \boldsymbol{n}$ 矩阵是一个子空间 $\mathbb{R}^m$. 注意 $\operatorname{Col} A$ 等于 $\mathbb{R}^m$ 只有当列 $A$ 跨度 $\mathbb{R}^m$. 否则， $\operatorname{Col} A$ 只是一部分 $\mathbb{R}^m$. $A$.

$$A(\mathbf{u}+\mathbf{v})=A \mathbf{u}+A \mathbf{v}=\mathbf{0}+\mathbf{0}=\mathbf{0}$$

## 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|线性代数代写linear algebra代考|MATH1051

statistics-lab™ 为您的留学生涯保驾护航 在代写线性代数linear algebra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。我们的专家在代写线性代数linear algebra代写方面经验极为丰富，各种代写线性代数linear algebra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|线性代数代写linear algebra代考|Perspective Projections

A three-dimensional object is represented on the two-dimensional computer screen by projecting the object onto a viewing plane. (We ignore other important steps, such as selecting the portion of the viewing plane to display on the screen.) For simplicity, let the $x y$-plane represent the computer screen, and imagine that the eye of a viewer is along the positive $z$-axis, at a point $(0,0, d)$. A perspective projection maps each point $(x, y, z)$ onto an image point $\left(x^, y^, 0\right)$ so that the two points and the eye position, called the center of projection, are on a line. See Figure 6(a).

The triangle in the $x z$-plane in Figure 6(a) is redrawn in part (b) showing the lengths of line segments. Similar triangles show that
$$\frac{x^}{d}=\frac{x}{d-z} \quad \text { and } \quad x^=\frac{d x}{d-z}=\frac{x}{1-z / d}$$
Similarly,
$$y^*=\frac{y}{1-z / d}$$
Using homogeneous coordinates, we can represent the perspective projection by a matrix, say, $P$. We want $(x, y, z, 1)$ to map into $\left(\frac{x}{1-z / d}, \frac{y}{1-z / d}, 0,1\right)$. Scaling these coordinates by $1-z / d$, we can also use $(x, y, 0,1-z / d)$ as homogeneous coordinates for the image. Now it is easy to display $P$. In fact,
$$P\left[\begin{array}{l} x \ y \ z \ 1 \end{array}\right]=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & -1 / d & 1 \end{array}\right]\left[\begin{array}{c} x \ y \ z \ 1 \end{array}\right]=\left[\begin{array}{c} x \ y \ 0 \ 1-z / d \end{array}\right]$$
EXAMPLE 8 Let $S$ be the box with vertices $(3,1,5),(5,1,5),(5,0,5),(3,0,5)$, $(3,1,4),(5,1,4),(5,0,4)$, and $(3,0,4)$. Find the image of $S$ under the perspective projection with center of projection at $(0,0,10)$.

## 数学代写|线性代数代写linear algebra代考|Column Space and Null Space of a Matrix

Subspaces of $\mathbb{R}^n$ usually occur in applications and theory in one of two ways. In both cases, the subspace can be related to a matrix.
The column space of a matrix $A$ is the set $\operatorname{Col} A$ of all linear combinations of the columns of $A$.
If $A=\left[\begin{array}{lll}\mathbf{a}_1 & \cdots & \mathbf{a}_n\end{array}\right]$, with the columns in $\mathbb{R}^m$, then $\operatorname{Col} A$ is the same as Span $\left{\mathbf{a}_1, \ldots, \mathbf{a}_n\right}$. Example 4 shows that the column space of an $\boldsymbol{m} \times \boldsymbol{n}$ matrix is a subspace of $\mathbb{R}^m$. Note that $\operatorname{Col} A$ equals $\mathbb{R}^m$ only when the columns of $A$ span $\mathbb{R}^m$. Otherwise, $\operatorname{Col} A$ is only part of $\mathbb{R}^m$.

EXAMPLE 4 Let $A=\left[\begin{array}{rrr}1 & -3 & -4 \ -4 & 6 & -2 \ -3 & 7 & 6\end{array}\right]$ and $\mathbf{b}=\left[\begin{array}{r}3 \ 3 \ -4\end{array}\right]$. Determine whether $\mathbf{b}$ is in the column space of $A$.

SOLUTION The vector $\mathbf{b}$ is a linear combination of the columns of $A$ if and only if $\mathbf{b}$ can be written as $A \mathbf{x}$ for some $\mathbf{x}$, that is, if and only if the equation $A \mathbf{x}=\mathbf{b}$ has a solution. Row reducing the augmented matrix $\left[A \begin{array}{ll}A & \mathbf{b}\end{array}\right]$,
$$\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ -4 & 6 & -2 & 3 \ -3 & 7 & 6 & -4 \end{array}\right] \sim\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ 0 & -6 & -18 & 15 \ 0 & -2 & -6 & 5 \end{array}\right] \sim\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ 0 & -6 & -18 & 15 \ 0 & 0 & 0 & 0 \end{array}\right]$$
we conclude that $A \mathbf{x}=\mathbf{b}$ is consistent and $\mathbf{b}$ is in $\operatorname{Col} A$.

The solution of Example 4 shows that when a system of linear equations is written in the form $A \mathbf{x}=\mathbf{b}$, the column space of $A$ is the set of all $\mathbf{b}$ for which the system has a solution.
The null space of a matrix $A$ is the set $\mathrm{Nul} A$ of all solutions of the homogeneous equation $A \mathbf{x}=\mathbf{0}$
When $A$ has $n$ columns, the solutions of $A \mathbf{x}=\mathbf{0}$ belong to $\mathbb{R}^n$, and the null space of $A$ is a subset of $\mathbb{R}^n$. In fact, $\mathrm{Nul} A$ has the properties of a subspace of $\mathbb{R}^n$.
The null space of an $m \times n$ matrix $A$ is a subspace of $\mathbb{R}^n$. Equivalently, the set of all solutions of a system $A \mathbf{x}=\mathbf{0}$ of $m$ homogeneous linear equations in $n$ unknowns is a subspace of $\mathbb{R}^n$.
PROOF The zero vector is in $\operatorname{Nul} A$ (because $A 0=0$ ). To show that $\mathrm{Nul} A$ satisfies the other two properties required for a subspace, take any $\mathbf{u}$ and $\mathbf{v}$ in $\mathrm{Nul} A$. That is, suppose $A \mathbf{u}=\mathbf{0}$ and $A \mathbf{v}=\mathbf{0}$. Then, by a property of matrix multiplication,
$$A(\mathbf{u}+\mathbf{v})=A \mathbf{u}+A \mathbf{v}=\mathbf{0}+\mathbf{0}=\mathbf{0}$$
Thus $\mathbf{u}+\mathbf{v}$ satisfies $A \mathbf{x}=\mathbf{0}$, and so $\mathbf{u}+\mathbf{v}$ is in $\operatorname{Nul} A$. Also, for any scalar $c, A(c \mathbf{u})=$ $c(A \mathbf{u})=c(0)=\mathbf{0}$, which shows that $c \mathbf{u}$ is in $\mathrm{Nul} A$.

To test whether a given vector $\mathbf{v}$ is in $\operatorname{Nul} A$, just compute $A \mathbf{v}$ to see whether $A \mathbf{v}$ is the zero vector. Because $\mathrm{Nul} A$ is described by a condition that must be checked for each vector, we say that the null space is defined implicitly. In contrast, the column space is defined explicitly, because vectors in Col A can be constructed (by linear combinations) from the columns of $A$. To create an explicit description of $\mathrm{Nul} A$, solve the equation $A \mathbf{x}=\mathbf{0}$ and write the solution in parametric vector form. (See Example 6 , below.) ${ }^2$

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Perspective Projections

y^*=\frac{y}{1-z / d}
$$使用齐次坐标，我们可以用矩阵表示透视投影，比如说， P. 我们想要 (x, y, z, 1) 映射到 \left(\frac{x}{1-z / d}, \frac{y}{1-z / d}, 0,1\right). 缩放这些坐标 1-z / d ，我们也可以使用 (x, y, 0,1-z / d) 作为图像的齐次坐 标。现在很容易显示 P. 实际上，$$
P[x y z 1]=\left[\begin{array}{llllllllllllllll}
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 / d & 1
\end{array}\right]\left[\begin{array}{ll}
x y z & -1
\end{array}\right]=\left[\begin{array}{lll}
x & 0 & 1
\end{array}\right.
$$例 8 让 S 是有顶点的盒子 (3,1,5),(5,1,5),(5,0,5),(3,0,5) ，(3,1,4),(5,1,4),(5,0,4) ， 和 (3,0,4). 找到图像 S 在投影中心位于的透视投影下 (0,0,10). ## 数学代写|线性代数代写linear algebra代考|Column Space and Null Space of a Matrix 通过将物体投影到观察平面上，三维物体在二维计算机屏幕上呈现。（我们忽略其他重要步㡜，例如选择 要在屏幕上显示的视图平面部分。) 为简单起见，让 x y-plane 代表计算机屏幕，并想象观众的眼睛沿着 正面 z-轴，在一点 (0,0, d). 透视投影映射每个点 (x, y, z) 到图像点 \ V \mathrm{eft}\left(\mathrm{X}^{\wedge}, y^{\wedge} ， 0 \backslash r i g h t\right) \$$ 上，这样两 个点和眼睛位置 (称为投影中心) 在一条线上。见图 6(a)。

$\left\langle f r a c\left{x^{\wedge}\right}{d}=|f r a c{x}{d z} \backslash q u a d| t e x t{\right.$ 和 $\left.} \backslash q u a d x^{\wedge}=\right| f r a c{d x}{d z}=\backslash f r a c{x}{1-z / d}$

$$y^*=\frac{y}{1-z / d}$$

$$P[x y z 1]=\left[\begin{array}{llllllllllllllll} 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 / d & 1 \end{array}\right]\left[\begin{array}{ll} x y z & -1 \end{array}\right]=\left[\begin{array}{lll} x & 0 & 1 \end{array}\right.$$

## 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|线性代数代写linear algebra代考|MATH1014

statistics-lab™ 为您的留学生涯保驾护航 在代写线性代数linear algebra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。我们的专家在代写线性代数linear algebra代写方面经验极为丰富，各种代写线性代数linear algebra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|线性代数代写linear algebra代考|APPLICATIONS TO COMPUTER GRAPHICS

Computer graphics are images displayed or animated on a computer screen. Applications of computer graphics are widespread and growing rapidly. For instance, computeraided design (CAD) is an integral part of many engineering processes, such as the aircraft design process described in the chapter introduction. The entertainment industry has made the most spectacular use of computer graphics – from the special effects in Amazing Spider-Man 2 to PlayStation 4 and Xbox One.

Most interactive computer software for business and industry makes use of computer graphics in the screen displays and for other functions, such as graphical display of data, desktop publishing, and slide production for commercial and educational presentations. Consequently, anyone studying a computer language invariably spends time learning how to use at least two-dimensional (2D) graphics.

This section examines some of the basic mathematics used to manipulate and display graphical images such as a wire-frame model of an airplane. Such an image (or picture) consists of a number of points, connecting lines or curves, and information about how to fill in closed regions bounded by the lines and curves. Often, curved lines are approximated by short straight-line segments, and a figure is defined mathematically by a list of points.

Among the simplest 2D graphics symbols are letters used for labels on the screen. Some letters are stored as wire-frame objects; others that have curved portions are stored with additional mathematical formulas for the curves.

EXAMPLE 1 The capital letter $\mathrm{N}$ in Figure 1 is determined by eight points, or vertices. The coordinates of the points can be stored in a data matrix, $D$.
$x$-coordinate $\left[\begin{array}{cccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \ 0 & .5 & .5 & 6 & 6 & 5.5 & 5.5 & 0 \ 0 & 0 & 6.42 & 0 & 8 & 8 & 1.58 & 8\end{array}\right]=D$
In addition to $D$, it is necessary to specify which vertices are connected by lines, but we omit this detail.

The main reason graphical objects are described by collections of straight-line segments is that the standard transformations in computer graphics map line segments onto other line segments. (For instance, see Exercise 27 in Section 1.8.) Once the vertices that describe an object have been transformed, their images can be connected with the appropriate straight lines to produce the complete image of the original object.

## 数学代写|线性代数代写linear algebra代考|Homogeneous 3D Coordinates

By analogy with the $2 \mathrm{D}$ case, we say that $(x, y, z, 1)$ are homogeneous coordinates for the point $(x, y, z)$ in $\mathbb{R}^3$. In general, $(X, Y, Z, H)$ are homogeneous coordinates for $(x, y, z)$ if $H \neq 0$ and
$$x=\frac{X}{H}, \quad y=\frac{Y}{H}, \quad \text { and } \quad z=\frac{Z}{H}$$
Each nonzero scalar multiple of $(x, y, z, 1)$ gives a set of homogeneous coordinates for $(x, y, z)$. For instance, both $(10,-6,14,2)$ and $(-15,9,-21,-3)$ are homogeneous coordinates for $(5,-3,7)$.

The next example illustrates the transformations used in molecular modeling to move a drug into a protein molecule.
EXAMPLE 7 Give $4 \times 4$ matrices for the following transformations:
a. Rotation about the $y$-axis through an angle of $30^{\circ}$. (By convention, a positive angle is the counterclockwise direction when looking toward the origin from the positive half of the axis of rotation-in this case, the $y$-axis.)
b. Translation by the vector $\mathbf{p}=(-6,4,5)$.
SOLUTION
a. First, construct the $3 \times 3$ matrix for the rotation. The vector $\mathbf{e}_1$ rotates down toward the negative $z$-axis, stopping at $\left(\cos 30^{\circ}, 0,-\sin 30^{\circ}\right)=(\sqrt{3} / 2,0,-.5)$. The vector $\mathbf{e}_2$ on the $y$-axis does not move, but $\mathbf{e}_3$ on the $z$-axis rotates down toward the positive $x$-axis, stopping at $\left(\sin 30^{\circ}, 0, \cos 30^{\circ}\right)=(.5,0, \sqrt{3} / 2)$. See Figure 5. From Section $1.9$, the standard matrix for this rotation is
$$\left[\begin{array}{ccc} \sqrt{3} / 2 & 0 & .5 \ 0 & 1 & 0 \ -.5 & 0 & \sqrt{3} / 2 \end{array}\right]$$
So the rotation matrix for homogeneous coordinates is
$$A=\left[\begin{array}{cccc} \sqrt{3} / 2 & 0 & .5 & 0 \ 0 & 1 & 0 & 0 \ -.5 & 0 & \sqrt{3} / 2 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$
b. We want $(x, y, z, 1)$ to map to $(x-6, y+4, z+5,1)$. The matrix that does this is
$$\left[\begin{array}{rrrr} 1 & 0 & 0 & -6 \ 0 & 1 & 0 & 4 \ 0 & 0 & 1 & 5 \ 0 & 0 & 0 & 1 \end{array}\right]$$

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|APPLICATIONS TO COMPUTER GRAPHICS

$\left[\begin{array}{llllllllllllllllllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 0 & .5 & .5 & 6 & 6 & 5.5 & 5.5 & 0 & 0 & 0 & 6.42 & 0 & 8 & 8 & 1.58 & 8\end{array}\right]$ 此外 $D$ ，有必要指定哪些顶点由线连接，但我们省略了这个细节。

## 数学代写|线性代数代写linear algebra代考|Homogeneous 3D Coordinates

$$x=\frac{X}{H}, \quad y=\frac{Y}{H}, \quad \text { and } \quad z=\frac{Z}{H}$$

a。旋转关于 $y$-轴通过一个角度 $30^{\circ}$. (按照惯例，正角是从旋转轴的正半边看原点时的逆时针方向一一在 这种情况下， $y$-轴。)
$\mathrm{b}$ 。向量翻译 $\mathbf{p}=(-6,4,5)$.

$\left(\cos 30^{\circ}, 0,-\sin 30^{\circ}\right)=(\sqrt{3} / 2,0,-.5)$. 载体 $\mathbf{e}_2$ 在 $y$-轴不移动，但 $\mathbf{e}_3$ 在 $z$-axis 向下旋转到正 $x$ 轴，停在 $\left(\sin 30^{\circ}, 0, \cos 30^{\circ}\right)=(.5,0, \sqrt{3} / 2)$. 参见图 5。从部分 $1.9$ ，这个旋转的标准矩阵是

b. 我们想要 $(x, y, z, 1)$ 映射到 $(x-6, y+4, z+5,1)$. 这样做的矩阵是

## 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|黎曼曲面代写Riemann surface代考|MTH3022

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface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。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface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|黎曼曲面代写Riemann surface代考|THE MAIN LEMMA

In this section we will count the cells in the chains $\varphi, \tau$, and $\psi$ that were defined in the previous section. Note that
$$\begin{gathered} \varphi=\sum_r(A K)^r \eta \ \tau=\sum_r(-K A)^r H \varphi \ \psi=\sum_r(-K A)^r K \nu \end{gathered}$$
We will show that the number of nondegenerate cubical cells in one of these chains is bounded by $C^n$, by parametrizing the cells with trees.

Suppose $z$ is a point in $Z_I$, with $|I|=n$. Consider the chain $F(K A)^r z$. It is a sum of cells of the form
$$F K_{k_r} \alpha_r \ldots K_{k_1} \alpha_1 z$$
Each of these cells is an $r$-cube. Our main construction will be to describe the points in these cells using graphs (which are trees).

Fix a sequence $\alpha_1, \ldots, \alpha_r$. In particular there is a sequence of indices $I=$ $I_0, I_1, \ldots, I_r$ such that $\alpha_j: I_{j-1} \rightarrow I_j$. We will associate a graph to this choice as follows. The vertices are arrayed in $r+1$ rows, with the top row having $n+2$ vertices and bottom row having $n+r+2$ vertices. The $j$ th row from the top has $n+j+2$ vertices. The vertices are numbered from right to left in each row, beginning with 0 , and we denote the $k$ th vertex in the $j$ th row by $v_{j k}$. The vertices at the ends of the rows, $v_{j 0}$ and $v_{j(n+j+1)}$, are called side vertices. The edges of the graph go from vertices in one row to vertices in the next. There is an edge connecting $v_{j-1 i}$ to $v_{j k}$ if and only if $\alpha_j^{+}(k)=i$. Thus in each row except the bottom one, there is exactly one vertex with two edges emanating from below, and all of the other vertices have one edge below. The edges, when drawn as straight lines, do not intersect, because the maps $\alpha^{+}$ are order preserving. The edges drawn from $v_{j 0}$ to $v_{j+10}$ and from $v_{j(n+j+1)}$ to $v_{(j+1)(n+j+2)}$ are called side edges.

## 数学代写|黎曼曲面代写Riemann surface代考|THE MAIN LEMMA

We can decompose the graph into strands, with the strands joining forks. The forks are the vertices which are connected to three edges (in other words the vertices $v_{j k}$ such that $\alpha_{j+1}^{+}(k)=\alpha_{j+1}^{+}(k+1)=k$ ), as well as, by convention, the top and bottom vertices. The strands are the unbroken sequences of edges joining forks, in other words the sequences of edges which meet at interior vertices with only two edges. Side strands are those consisting of side edges. The graph formed by the forks and strands considered as vertices and edges, is a union of binary trees. If a number is assigned to each non-side edge, then one obtains a number for each non-side strand as follows. Suppose $\sigma$ is a strand, composed of edges $e_1, \ldots, e_m$. Set
$$t(\sigma)=\min \left(1, t\left(e_1\right)+\ldots+t\left(e_m\right)\right) .$$
In the above construction, the point $u$ depends only on the numbers $t(\sigma)$ assigned to the strands. Here is another description of the construction of $u$. For each strand $\sigma$ there are indices $i(\sigma)$ and $j(\sigma)$, representing the indices corresponding to the left and right sides of the edges in the strand, respectively. If the strand $\sigma$ contains an edge ending in a vertex $v_{j k}$, then $i(\sigma)=i_{j, k-1}$ and $j(\sigma)=i_{j, k}$. (The notation $i(e)$ and $j(e)$ will also be used for an edge $e$.) Realize the tree geometrically, with a strand $\sigma$ represented by a line segment of length 1. Let $T$ denote the geometric realization of the tree. Then the function $t$ from the set of strands into $[0,1]$, and the initial point $z$, determine a map $\Psi_{z, t}: T \rightarrow Z$. Write $z=\left(z_1, \ldots, z_n\right)$. The top vertices of the tree go to the points $z_k \in Z$. The left and right side strands are mapped to $P$ and $Q$ respectively. If $\sigma$ is any strand, $\Psi_{z, t}$ maps the segment corresponding to $\sigma$ into $Z$ using the flow $f_{i(\sigma) j(\sigma)}$, beginning with the point corresponding to the fork $v$ at the top of $\sigma$, and moving at speed $t(\sigma)$. The beginning point $\Psi_{z, t}(v)$ has already been constructed inductively. If $p$ is a point on the segment $\sigma$, at distance $y$ below the fork $v, \Psi_{z, t}(p)=f_{i(\sigma) j(\sigma)}\left(\Psi_{z, t}(v), t(\sigma) y\right)$. Finally, the the values of $\Psi_{z, t}$ on the $n+r$ bottom vertices provide the points $u_1, \ldots, u_{n+r}$ to determine $u=u(z, t) \in Z_{I_r}$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|THE MAIN LEMMA

$$\varphi=\sum_r(A K)^r \eta \tau=\sum_r(-K A)^r H \varphi \psi=\sum_r(-K A)^r K \nu$$

$$F K_{k_r} \alpha_r \ldots K_{k_1} \alpha_1 z$$

## 数学代写|黎曼曲面代写Riemann surface代考|THE MAIN LEMMA

$$t(\sigma)=\min \left(1, t\left(e_1\right)+\ldots+t\left(e_m\right)\right) .$$

## 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。