## 数学代写|线性代数代写linear algebra代考|MAST10007

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|Euler’s Number and Natural Logarithms

There is a special number that shows up quite a bit in math called Euler’s number $e$. It is a special number much like Pi $\pi$ and is approximately 2.71828. $e$ is used a lot because it mathematically simplifies a lot of problems. We will cover $e$ in the context of exponents and logarithms.

Back in high school, my calculus teacher demonstrated Euler’s number in several exponential problems. Finally I asked, “Mr. Nowe, what is $e$ anyway? Where does it come from?” I remember never being fully satisfied with the explanations involving rabbit populations and other natural phenomena. I hope to give a more satisfying explanation here.
Why Euler’s Number Is Used So Much
A property of Euler’s number is its exponential function is a derivative to itself, which is convenient for exponential and logarithmic functions. We will learn about derivatives later in this chapter. In many applications where the base does not really matter, we pick the one that results in the simplest derivative, and that is Euler’s number. That is also why it is the default base in many data science functions.

Here is how I like to discover Euler’s number. Let’s say you loan $\$ 100$to somebody with$20 \%$interest annually. Typically, interest will be compounded monthly, so the interest each month would be$.20 / 12=.01666$. How much will the loan balance be after two years? To keep it simple, let’s assume the loan does not require payments (and no payments are made) until the end of those two years. Putting together the exponent concepts we learned so far (or perhaps pulling out a finance textbook), we can come up with a formula to calculate interest. It consists of a balance$A$for a starting investment$P$, interest rate$r$, time span$t$(number of years), and periods$n$(number of months in each year). Here is the formula: $$A=P \times\left(1+\frac{r}{n}\right)^{n t}$$ So if we were to compound interest every month, the loan would grow to$\$148.69$ as calculated here:
$$A=P \times\left(1+\frac{r}{n}\right)^{n t}$$

$$100 \times\left(1+\frac{.20}{12}\right)^{12 \times 2}=148.6914618$$
If you want to do this in Python, try it out with the code in Example 1-13.

## 数学代写|线性代数代写linear algebra代考|Natural Logarithms

When we use $e$ as our base for a logarithm, we call it a natural logarithm. Depending on the platform, we may use $\ln ()$ instead of $\log ()$ to specify a natural logarithm. So rather than express a natural logarithm expressed as $\log {e} 10$ to find the power raised on $e$ to get 10 , we would shorthand it as $\ln (10)$ : $$\log {e} 10=\ln (10)$$
However, in Python, a natural logarithm is specified by the log() function. As discussed earlier, the default base for the $\log ()$ function is $e$. Just leave the second argument for the base empty and it will default to using $e$ as the base shown in Example 1-15.
Example 1-15. Calculating the natural logarithm of 10 in Python
from nath import loge raised to what power gives us 10 ?

$x=\log (10)$ We will use $e$ in a number of places throughout this book. Feel free to experiment with exponents and logarithms using Excel, Python, Desmos.com, or any other calculation platform of your choice. Make graphs and get comfortable with what these functions look like.

## 数学代写|线性代数代写linear algebra代考|Euler’s Number and Natural Logarithms

$$\begin{gathered} A=P \times\left(1+\frac{r}{n}\right)^{n t} \ 100 \times\left(1+\frac{.20}{12}\right)^{12 \times 2}=148.6914618 \end{gathered}$$

## 数学代写|线性代数代写linear algebra代考|Natural Logarithms

$$\log e 10=\ln (10)$$

$x=\log (10)$ 我们将使用 $e$ 在本书的许多地方。随意使用 Excel、Python、Desmos.com 或您选择的任何其他计 算平台来试验指数和对数。制作图表并熟悉这些函数的外观。

## 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|MTH 2106

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|Summations

I promised not to use equations full of Greek symbols in this book. However, there is one that is so common and useful that I would be remiss to not cover it. A summation is expressed as a sigma $\Sigma$ and adds elements together.

For example, if I want to iterate the numbers 1 through 5 , multiply each by 2 , and sum them, here is how I would express that using a summation. Example 1-10 shows how to execute this in Python.
$$\sum_{i=1}^{5} 2 i=(2) 1+(2) 2+(2) 3+(2) 4+(2) 5=30$$
Example 1-10. Performing a summation in Python
summation $=\operatorname{sum}(2 * i$ for i in range $(1,6))$
print(summation)
Note that $i$ is a placeholder variable representing each consecutive index value we are iterating in the loop, which we multiply by 2 and then sum all together. When you are iterating data, you may see variables like $x_{i}$ indicating an element in a collection at index $i$.

It is also common to see $n$ represent the number of items in a collection, like the number of records in a dataset. Here is one such example where we iterate a collection of numbers of size $n$, multiply each one by 10 , and sum them:
$$\sum_{i=1}^{n} 10 x_{i}$$
In Example 1-11 we use Python to execute this expression on a collection of four numbers. Note that in Python (and most programming languages in general) we typically reference items starting at index 0 , while in math we start at index 1 . Therefore, we shift accordingly in our iteration by starting at 0 in our range().

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

Exponents multiply a number by itself a specified number of times. When you raise 2 to the third power (expressed as $2^{3}$ using 3 as a superscript), that is multiplying three 2s together:
$$2^{3}=2 * 2 * 2=8$$
The base is the variable or value we are exponentiating, and the exponent is the number of times we multiply the base value. For the expression $2^{3}, 2$ is the base and 3 is the exponent.

Exponents have a few interesting properties. Say we multiplied $x^{2}$ and $x^{3}$ together. Observe what happens next when I expand the exponents with simple multiplication and then consolidate into a single exponent:
$$x^{2} x^{3}=\left(x^{} x\right)^{}\left(x^{} x^{} x\right)=x^{2+3}=x^{5}$$
When we multiply exponents together with the same base, we simply add the exponents, which is known as the product rule. Let me emphasize that the base of all multiplied exponents must be the same for the product rule to apply.
Let’s explore division next. What happens when we divide $x^{2}$ by $x^{5}$ ?
$$\frac{x^{2}}{x^{5}}$$ \begin{aligned} &\frac{x^{} x}{x^{} x^{} x^{} x^{} x} \ &\frac{1}{x^{} x^{*} x} \ &\frac{1}{x^{3}}=x^{-3} \end{aligned}
As you can see, when we divide $x^{2}$ by $x^{5}$ we can cancel out two $x^{\prime}$ ‘ in the numerator and denominator, leaving us with $\frac{1}{x^{3}}$. When a factor exists in both the numerator and denominator, we can cancel out that factor.

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

$$\sum_{i=1}^{5} 2 i=(2) 1+(2) 2+(2) 3+(2) 4+(2) 5=30$$

summation中执行求和 $=\operatorname{sum}(2 * i$ 因为我在范围内 $(1,6))$

$$\sum_{i=1}^{n} 10 x_{i}$$

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

$$2^{3}=2 * 2 * 2=8$$

$$x^{2} x^{3}=(x x)(x x x)=x^{2+3}=x^{5}$$

$$\begin{gathered} \frac{x^{2}}{x^{5}} \ \frac{x x}{x x x x x} \quad \frac{1}{x x^{*} x} \frac{1}{x^{3}}=x^{-3} \end{gathered}$$

## 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|МATH 1014

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|Basic Math and Calculus Review

We will kick off the first chapter covering what numbers are and how variables and functions work on a Cartesian system. We will then cover exponents and logarithms. After that, we will learn the two basic operations of calculus: derivatives and integrals.
Before we dive into the applied areas of essential math such as probability, linear algebra, statistics, and machine learning, we should probably review a few basic math and calculus concepts. Before you drop this book and run screaming, do not worry! I will present how to calculate derivatives and integrals for a function in a way you were probably not taught in college. We have Python on our side, not a pencil and paper. Even if you are not familiar with derivatives and integrals, you still do not need to worry.

I will make these topics as tight and practical as possible, focusing only on what will help us in later chapters and what falls under the “essential math” umbrella.

This is by no means a comprehensive review of high school and college math. If you want that, a great book to check out is No Bullshit Guide to Math and Physics by Ivan Savov (pardon my French). The first few chapters contain the best crash course on high school and college math I have ever seen. The book Mathematics 1001 by Dr. Richard Elwes has some great content as well, and in bite-sized explanations.

## 数学代写|线性代数代写linear algebra代考|Number Theory

What are numbers? I promise to not be too philosophical in this book, but are numbers not a construct we have defined? Why do we have the digits 0 through 9 , and not have more digits than that? Why do we have fractions and decimals and not just whole numbers? This area of math where we muse about numbers and why we designed them a certain way is known as number theory.

Number theory goes all the way back to ancient times, when mathematicians studied different number systems, and it explains why we have accepted them the way we do today. Here are different number systems that you may recognize:
Natural numbers
These are the numbers $1,2,3,4,5 \ldots$ and so on. Only positive numbers are included here, and they are the earliest known system. Natural numbers are so ancient cavemen scratched tally marks on bones and cave walls to keep records.
Whole numbers
Adding to natural numbers, the concept of ” 0 ” was later accepted; we call these “whole numbers.” The Babylonians also developed the useful idea for place-holding notation for empty “columns” on numbers greater than 9 , such as “10,” “ 1,000 ,” or “1,090.” Those zeros indicate no value occupying that column.
Integers
Integers include positive and negative natural numbers as well as 0 . We may take them for granted, but ancient mathematicians deeply distrusted the idea of negative numbers. But when you subtract 5 from 3, you get $-2$. This is useful especially when it comes to finances where we measure profits and losses. In $628 \mathrm{AD}$, an Indian mathematician named Brahmagupta showed why negative numbers were necessary for arithmetic to progress with the quadratic formula, and therefore integers became accepted.

## 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|MATH1012

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|Linear Combinations

Suppose we are working within a subspace for which all the radiographs satisfy a particular (significant) property, call it property $S$. This means that the subspace is defined as the set of all radiographs with property $S$. Because the subspace is not trivial (that is, it contains more than just the zero radiograph) it consists of an infinite number of radiographs. Suppose also that we have a handful of radiographs

that we know are in this subspace, but then a colleague brings us a new radiograph, $r$, one with which we have no experience and the colleague needs to know whether $r$ has property $S$. Since the set of radiographs defined by property $S$ is a subspace, we can perform a quick check to see if $r$ can be formed from those radiographs with which we are familiar, using arithmetic operations. If we find the answer to this question is “yes,” then we know $r$ has property $S .$ We know this because subspaces are closed under scalar multiplication and vector addition. If we find the answer to be “no, we still have more work to do. We cannot yet conclude whether or not $r$ has property $S$ because there may be radiographs with property $S$ that are still unknown to us.

We have also been exploring one-dimensional heat states on a finite interval. We have seen that the subset of heat states with fixed (zero) endpoint temperature differential is a subspace of the vector space of heat states. The collection of vectors in this subspace is relatively easy to identify: finitevalued and zero at the ends. However, if a particular heat state on a rod could cause issues with future functioning of a diffusion welder, an engineer might be interested in whether the subspace of possible heat states might contain this detrimental heat state. We may wish to determine if one such heat state is an arithmetic combination of several others.

In Section 3.1.1, we introduce the terminology of linear combinations for describing when a vector can be formed from a finite number of arithmetic operations on a specified set of vectors. In Sections $3.1 .3$ and 3.1.4 we consider linear combinations of vectors in Euclidean space ( $\left.\mathbb{R}^{n}\right)$ and connect such linear combinations to the inhomogeneous and homogeneous matrix equations $A x=b$ and $A x=0$, respectively. Finally, in Section 3.1.5, we discuss the connection between inhomogeneous and homogeneous systems.

## 数学代写|线性代数代写linear algebra代考|Linear Combinations

We now assign terminology to describe vectors that have been created from (a finite number of) arithmetic operations with a specified set of vectors.

Let $(V,+, \cdot)$ be a vector space over $\mathbb{F}$. Given a finite set of vectors $v_{1}, v_{2}, \cdots, v_{k} \in V$, we say that the vector $w \in V$ is a linear combination of $v_{1}, v_{2}, \cdots, v_{k}$ if $w=a_{1} v_{1}+a_{2} v_{2}+\cdots+a_{k} v_{k}$ for some scalar coefficients $a_{1}, a_{2}, \cdots, a_{k} \in \mathbb{F}$.

Corollary 2.5.4 says that a subspace is a nonempty subset of a vector space that is closed under scalar multiplication and vector addition. Using this new terminology, we can say that a subspace is closed under linear combinations.
Following is an example in the vector space $\mathcal{I}_{4 \times 4}$ of $4 \times 4$ images.
Example 3.1.2 Consider the $4 \times 4$ grayscale images from page 11. Image 2 is a linear combination of Images $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ with scalar coefficients $\frac{1}{2}, 0$, and 1 , respectively, because

** Watch Your Language! When communicating whether or not a vector can be written as a linear combination of other vectors, you should recognize that the term “linear combination” is a property applied to vectors, not sets. So, we make statements such as
$\checkmark w$ is a linear combination of $v_{1}, v_{2}, v_{3}$
$\checkmark w$ is not a linear combination of $u_{1}, u_{2}, u_{3}, \ldots u_{n} \cdot$
$\checkmark w$ is a linear combination of vectors in $U=\left{v_{1}, v_{2}, v_{3}\right}$.
We do not say
$X w$ is a linear combination of $U$.
In the remainder of this section, we focus on the question: Can a given vector be written as a linear combination of vectors from some specified set of vectors?

## 数学代写|线性代数代写linear algebra代考|Matrix Products

In this section, we state definitions and give examples of matrix products. We expect that most linear algebra students have already learned to multiply matrices, but realize that some students may need a reminder. As with $\mathbb{R}, \mathcal{M}_{m \times n}$ can be equipped with other operations (besides addition and scalar multiplication) with which it is associated, namely matrix multiplication. But unlike matrix addition the matrix product is not defined component-wise.

Given an $n \times m$ matrix $A=\left(a_{i, j}\right)$ and an $m \times \ell$ matrix $B=\left(b_{i, j}\right)$, we define the matrix product of $A$ and $B$ to be the $n \times \ell$ matrix $A B=\left(c_{i, j}\right)$ where
$$c_{i, j}=\sum_{k=1}^{m} a_{i, k} b_{k, j}$$

We call this operation matrix multiplication.
Notation. There is no “.” between the matrices, rather they are written in juxtaposition to show a difference between the notation of a scalar product and the notation of a matrix product. The definition requires that the number of columns of $A$ is the same as the number of rows of $B$ for the product $A B$.
Example 3.1.8 Let
$$P=\left(\begin{array}{ll} 1 & 2 \ 3 & 4 \ 5 & 6 \end{array}\right), \quad Q=\left(\begin{array}{rr} 2 & 1 \ 1 & -1 \ 2 & 1 \end{array}\right), \text { and } R=\left(\begin{array}{rr} 2 & 0 \ 1 & -2 \end{array}\right)$$
Since both $P$ and $Q$ are $3 \times 2$ matrices, we see that the number of columns of $P$ is not the same as the number of rows of $Q$. Thus, $P Q$ is not defined. But, since $P$ has 2 columns and $R$ has 2 rows, $P R$ is defined. Let’s compute the matrix product $P R$. We can compute each entry as in Definition 3.1.7.
Position Computation
$$\begin{array}{ll} (i, j) & p_{i, 1} r_{1, j}+p_{i, 2} r_{2, j} \ \hline(1,1) & 1 \cdot 2+2 \cdot 1 \ (1,2) & 1 \cdot 0+2 \cdot(-2) \ (2,1) & 3 \cdot 2+4 \cdot 1 \ (2,2) & 3 \cdot 0+4 \cdot(-2) \ (3,1) & 5 \cdot 2+6 \cdot 1 \ (3,2) & 5 \cdot 0+6 \cdot(-2) \end{array}$$
Typically, when writing this out, we write it as
\begin{aligned} P R &=\left(\begin{array}{ll} 1 & 2 \ 3 & 4 \ 5 & 6 \end{array}\right)\left(\begin{array}{rr} 2 & 0 \ 1 & -2 \end{array}\right) \ &=\left(\begin{array}{rr} 1 \cdot 2+2 \cdot 1 & 1 \cdot 0+2 \cdot(-2) \ 3 \cdot 2+4 \cdot 1 & 3 \cdot 0+4 \cdot(-2) \ 5 \cdot 2+6 \cdot 1 & 5 \cdot 0+6 \cdot(-2) \end{array}\right) \ &=\left(\begin{array}{rr} 4 & -4 \ 10 & -8 \ 16 & -12 \end{array}\right) . \end{aligned}
In the above example, the result of the matrix product was a matrix of the same size as $P$. Let’s do another example to show that this is not always the case.

## 数学代写|线性代数代写linear algebra代考|Linear Combinations

** 注意你说的话！在交流一个向量是否可以写成其他向量的线性组合时，您应该认识到术语“线性组合”是应用于向量而不是集合的属性。所以，我们做这样的陈述
✓在是一个线性组合在1,在2,在3
✓在不是的线性组合在1,在2,在3,…在n⋅
✓在是向量的线性组合U=\left{v_{1}, v_{2}, v_{3}\right}U=\left{v_{1}, v_{2}, v_{3}\right}.

X在是一个线性组合在.

## 数学代写|线性代数代写linear algebra代考|Matrix Products

C一世,j=∑ķ=1米一个一世,ķbķ,j

\begin{array}{ll} (i, j) & p_{i, 1} r_{1, j}+p_{i, 2} r_{2, j} \ \hline(1,1) & 1 \ cdot 2+2 \cdot 1 \ (1,2) & 1 \cdot 0+2 \cdot(-2) \ (2,1) & 3 \cdot 2+4 \cdot 1 \ (2,2) & 3 \cdot 0+4 \cdot(-2) \ (3,1) & 5 \cdot 2+6 \cdot 1 \ (3,2) & 5 \cdot 0+6 \cdot(-2) \end{数组}\begin{array}{ll} (i, j) & p_{i, 1} r_{1, j}+p_{i, 2} r_{2, j} \ \hline(1,1) & 1 \ cdot 2+2 \cdot 1 \ (1,2) & 1 \cdot 0+2 \cdot(-2) \ (2,1) & 3 \cdot 2+4 \cdot 1 \ (2,2) & 3 \cdot 0+4 \cdot(-2) \ (3,1) & 5 \cdot 2+6 \cdot 1 \ (3,2) & 5 \cdot 0+6 \cdot(-2) \end{数组}

## 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|Subsets and Subspaces

Let $\left(V,+,{ }^{\circ}\right)$ be a vector space. In this section we discuss conditions on a subset of a vector space that will guarantee the subset is also a vector space. Recall that a subset of $V$ is a set that contains some of the elements of $V$. We define subset more precisely here.

Let $V$ and $W$ be sets. We say that $W$ is a subset of $V$ if every element of $W$ is an element of $V$ and we write $W \subset V$ or $W \subseteq V$. In the case where $W \neq V$ (there are elements of $V$ that are not in $W$ ), we say that $W$ is a proper subset of $V$ and we write $W \subseteq V$.

In a vector space context, we always assume the same operations on $W$ as we have defined on $V$.
Let $W$ be a subset of $V$. We are interested in subsets that also satisfy the vector space properties (recall Definition 2.3.5).

Let $(V,+, \cdot)$ be a vector space over a field $\mathbb{F}$. If $W \subseteq V$, then we say that $W$ is a subspace of $(V,+, \cdot)$ whenever $(W,+, \cdot)$ is also a vector space.

Now consider which vector space properties of $(V,+, \cdot)$ must also be true of the subset $W$. Which properties are not necessarily true? The commutative, associative, and distributive properties still hold because the operations are the same, the scalars come from the same scalar field, and elements of $W$ come from the set $V$. Therefore, since these properties are true in $V$, they are true in $W$. We say that these properties are inherited from $V$ since $V$ is like a parent set to $W$. Also, since, we do not change the scalar set when considering a subset, the scalar 1 is still an element of the scalar set. This tells us that we can determine whether a subset of a vector space is, itself, a vector space, by checking those properties that depend on how the subset differs from the parent vector space. The properties we need to check are the following
(P1) $W$ is closed under addition.
(P2) $W$ is closed under scalar multiplication.
(P8) $W$ contains the additive identity, denoted $0 .$
(P9) $W$ contains additive inverses.
With careful consideration, we see that, because $V$ contains additive inverses, then if (P1), (P2), and (P8) are true for $W$, it follows that $W$ must also contain additive inverses (see Exercise 14). Hence, as the following theorem states, we need only test for properties (P1), (P2), and (P8) in order to determine whether a subset is a subspace.

## 数学代写|线性代数代写linear algebra代考|Examples of Subspaces

Every vector space $(V,+, \cdot)$ has at least the following two subspaces.
Theorem $2.5 .7$
Let $(V,+, \cdot)$ be a vector space. Then $V$ is itself a subspace of $(V,+, \cdot)$.
Proof. Since every set is a subset of itself, the result follows from Definition $2.5 .2 .$
Theorem $2.5 .8$
Let $(V,+, \cdot)$ be a vector space. Then the set ${0}$ is a subspace of $(V,+, \cdot)$.
The proof is Exercise $19 .$
Example 2.5.9 Recall Example 2.4.9 from the last section. Let $V \subset \mathbb{R}^{3}$ be the set of all solutions to the equation $x_{1}+3 x_{2}-x_{3}=0$. Then $V$ is a subspace of $\mathbb{R}^{3}$, with the standard operations.

More generally, as we saw in the last section, the set of solutions to any homogeneous linear equation with $n$ variables is a subspace of $\left(\mathbb{R}^{n},+, \cdot\right)$.

Example 2.5.10 Consider the coordinate axes as a subset of the vector space $\mathbb{R}^{2}$. That is, let $T \subset \mathbb{R}^{2}$ be defined by
$$T=\left{x=\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2} \mid x_{1}=0 \text { or } x_{2}=0\right}$$
$T$ is not a subspace of $\left(\mathbb{R}^{2},+, \cdot\right)$, because although 0 is in $T$, making $T \neq \emptyset, T$ does not have the property that for all $x, y \in T$ and for all $\alpha, \beta \in \mathbb{R}, \alpha x+\beta y \in T$. To verify this, we need only produce one example of vectors $x, y \in T$ and scalars $\alpha, \beta \in \mathbb{R}$ so that $\alpha x+\beta y$ is not in $T$. Notice that $x=(0,1)$, $y=(1,0)$ are elements of $T$ and $\alpha=\beta=1$ are in $\mathbb{R}$. Since $1 \cdot x+1 \cdot y=(1,1)$ which is not in $T$, $T$ does not satisfy the subspace property.

Example 2.5.11 Consider $W=\left{(a, b, c) \in \mathbb{R}^{3} \mid c=0\right} . W$ is a subspace of $\mathbb{R}^{3}$, with the standard operations of addition and scalar multiplication. See Exercise $9 .$

## 数学代写|线性代数代写linear algebra代考|Building New Subspaces

In this section, we investigate the question, “If we start with two subspaces of the same vector space, to some observations that will simplify some previous examples and give us new tools for proving that subsets are subspaces. We first consider intersections and unions.
Definition $2.5 .17$
Let $S$ and $T$ be sets.

• The intersection of $S$ and $T$, written $S \cap T$, is the set containing all elements that are in both $S$ and $T$.
• The union of $S$ and $T$, written $S \cup T$, is the set containing all elements that are in either $S$ or $T$ (or both).
The intersection of two subspaces is a also a subspace.

Proof. Let $W_{1}$ and $W_{2}$ be subspaces of $(V,+, \cdot)$. We will show that the intersection of $W_{1}$ and $W_{2}$ is nonempty and closed under scalar multiplication and vector addition. To show that $W_{1} \cap W_{2}$ is nonempty, we notice that since both $W_{1}$ and $W_{2}$ contain the zero vector, so does $W_{1} \cap W_{2}$.

Now, let $u$ and $v$ be elements of $W_{1} \cap W_{2}$ and let $\alpha$ be a scalar. Since $W_{1}$ and $W_{2}$ are closed under addition and scalar multiplication, we know that $\alpha \cdot u+v$ is also in both $W_{1}$ and $W_{2}$. That is, $\alpha \cdot u+v$ is in $W_{1} \cap W_{2}$, so by Corollary $2.5 .6 W_{1} \cap W_{2}$ is closed under addition and scalar multiplication.
Thus, by Corollary $2.5 .4, W_{1} \cap W_{2}$ is a subspace of $(V,+, \cdot)$.
An important example involves solutions to homogeneous equations, which we first considered in Example 2.4.12.

Example 2.5.19 The solution set of a single homogeneous equation in $n$ variables is a subspace of $\mathbb{R}^{n}$ (see Example 2.4.12). By Theorem 2.5.18, the intersection of the solution sets of any $k$ homogeneous equations in $n$ variables is also subspace of $\mathbb{R}^{n}$.

In other words, if a system of linear equations consists only of homogeneous equations, then the set of solutions forms a subspace of $\mathbb{R}^{n}$. This is such an important result that we promote it from example to theorem.

## 数学代写|线性代数代写linear algebra代考|Subsets and Subspaces

（P1）在在添加下关闭。
(P2)在在标量乘法下是闭合的。
(P8)在包含加法身份，表示为0.
(P9)在包含加法逆元。

## 数学代写|线性代数代写linear algebra代考|Examples of Subspaces

T=\left{x=\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2} \mid x_{1}=0 \text { 或 } x_{2} =0\右}T=\left{x=\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2} \mid x_{1}=0 \text { 或 } x_{2} =0\右}

## 数学代写|线性代数代写linear algebra代考|Building New Subspaces

• 的交叉点小号和吨, 写小号∩吨, 是包含两者中所有元素的集合小号和吨.
• 工会小号和吨, 写小号∪吨, 是包含所有元素的集合小号或者吨（或两者）。
两个子空间的交集也是一个子空间。

## 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|MATH1071

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|Solution Spaces

In this section, we consider solution sets of linear equations. If an equation has $n$ variables, its solution set is a subset of $\mathbb{R}^{n}$. When is this set a vector space?
Example 2.4.9 Let $V \subseteq \mathbb{R}^{3}$ be the set of all solutions to the equation $x_{1}+3 x_{2}-x_{3}=0$. That is,
$$V=\left{\left(\begin{array}{l} x_{1} \ x_{2} \ x_{3} \end{array}\right) \in \mathbb{R}^{3} \mid x_{1}+3 x_{2}-x_{3}=0\right}$$
The set $V$ together with the operations $+$ and – inherited from $\mathbb{R}^{3}$ forms a vector space.
Proof. Let $V$ be the set defined above and let
$$u=\left(\begin{array}{l} u_{1} \ u_{2} \ u_{3} \end{array}\right), v=\left(\begin{array}{l} v_{1} \ v_{2} \ v_{3} \end{array}\right) \in V .$$
Let $\alpha \in \mathbb{R}$. Notice that Properties (P3)-(P7) and (P10) of Definition $2.3 .5$ only depend on the definition of addition and scalar multiplication on $\mathbb{R}^{3}$ and, therefore, are inherited properties from $\mathbb{R}^{3}$. Hence we need only check properties (P1), (P2), (P8), and (P9). Since $u, v \in V$, we know that

$$u_{1}+3 u_{2}-u_{3}=0 \text { and } v_{1}+3 v_{2}-v_{3}=0$$
We will use this result to show the closure properties.
First, notice that $u+v=\left(\begin{array}{l}u_{1}+v_{1} \ u_{2}+v_{2} \ u_{3}+v_{3}\end{array}\right)$. Notice also that
\begin{aligned} \left(u_{1}+v_{1}\right)+3\left(u_{2}+v_{2}\right)-\left(u_{3}+v_{3}\right) &=\left(u_{1}+3 u_{2}-u_{3}\right)+\left(v_{1}+3 v_{2}-v_{3}\right) \ &=0+0=0 \end{aligned}
Thus, $u+v \in V$. Since $u$ and $v$ are arbitrary vectors in $V$ it follows that $V$ is closed under addition.
Next, notice that $\alpha \cdot u=\left(\begin{array}{l}\alpha u_{1} \ \alpha u_{2} \ \alpha u_{3}\end{array}\right)$ and
\begin{aligned} \alpha u_{1}+3 \alpha u_{2}-\alpha u_{3} &=\alpha\left(u_{1}+3 u_{2}-u_{3}\right) \ &=\alpha(0)=0 \end{aligned}
Therefore, $\alpha u \in V$. Hence $V$ is closed under scalar multiplication.

## 数学代写|线性代数代写linear algebra代考|Other Vector Spaces

In many areas of mathematics, we learn about concepts that relate to vector spaces, though the details of vector space properties may be simply assumed or not well established. In this section, we look at some of these concepts and recognize how vector space properties are present.
Sequence Spaces
Here, we explore sequences such as those discussed in Calculus. We consider the set of all sequences in the context of vector space properties. First, we give a formal definition of sequences.

A sequence of real numbers is a function $s: \mathbb{N} \rightarrow \mathbb{R}$. That is, $s(n)=a_{n}$ for $n=1,2, \cdots$ where $a_{n} \in \mathbb{R}$. A sequence is denoted $\left{a_{n}\right}$. Let $\mathcal{S}(\mathbb{R})$ be the set of all sequences. Let $\left{a_{n}\right}$ and $\left{b_{n}\right}$ be sequences in $\mathcal{S}(\mathbb{R})$ and $\alpha$ in $\mathbb{R}$. Define sequence addition and scalar multiplication with a sequence by
$$\left{a_{n}\right}+\left{b_{n}\right}=\left{a_{n}+b_{n}\right} \text { and } \alpha \cdot\left{a_{n}\right}=\left{\alpha a_{n}\right} .$$
In Exercise 15, we show that $\mathcal{S}(\mathbb{R})$, with these (element-wise) operations, forms a vector space over $\mathbb{R}$.

Example 2.4.15 (Eventually Zero Sequences) Let $\mathcal{S}{\text {fin }}(\mathbb{R})$ be the set of all sequences that have a finite number of nonzero terms. Then $\mathcal{S}{\text {fin }}(\mathbb{R})$ is a vector space with operations as defined in Definition 2.4.14. (See Exercise 16.)

We find vector space properties for sequences to be very useful in the development of calculus concepts such as limits. For example, if we want to apply a limit to the sum of sequences, we need to know that the sum of two sequences is indeed a sequence. More of these concepts will be discussed later, after developing more linear algebra ideas.

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

PetPics, a pet photography company specializing in portraits, wants to post photos for clients to review, but to protect their artistic work, they only post electronic versions that have copyright text. The text is

added to all images, produced by the company, by overwriting, with zeros, in the appropriate pixels, as shown in Figure 2.18. Only pictures that have zeros in these pixels are considered legitimate images.
The company also wants to allow clients to make some adjustments to the pictures: the adjustments include brightening/darkening, and adding background or little figures like hearts, flowers, or squirrels. It turns out that these operations can all be accomplished by adding other legitimate images and multiplying by scalars, as defined in Section $2.3$.

It is certainly true that the set of all legitimate images of the company’s standard $(m \times n)$-pixel size is contained in the vector space $\mathcal{I}_{m \times n}$ of all $m \times n$ images, so we could mathematically work in this larger space. But, astute employees of the company who enjoy thinking about linear algebra notice that actually the set of legitimate images itself satisfies the 10 properties of a vector space. Specifically, adding any two images with the copyright text (for example, adding a squirrel to a portrait of a golden retriever) produces another image with the same copyright text, and multiplying an image with the copyright text by a scalar (say, to brighten it) still results in an image with the copyright text. Hence, it suffices to work with the smaller set of legitimate images.

In fact, very often the sets of objects that we want to focus on are actually only subsets of larger vector spaces, and it is useful to know when such a set forms a vector space separately from the larger vector space.
Here are some examples of subsets of vector spaces that we have encountered so far.

1. Solution sets of homogeneous linear equations, with $n$ variables, are subsets of $\mathbb{R}^{n}$.
2. Radiographs are images with nonnegative values and represent a subset of the larger vector space of images with the given geometry.
3. The set of even functions on $\mathbb{R}$ is a subset of the vector space of functions on $\mathbb{R}$.
4. Polynomials of order 3 form a subset of the vector space $\mathcal{P}_{5}(\mathbb{R})$.
5. Heat states on a rod in a diffusion welding process (the collection of which is $H_{m}(\mathbb{R})$ ) form a subset of all possible heat states because the temperature is fixed at the ends of the rod.
6. The set of sequences with exactly 10 nonzero terms is a subset of the set of sequences with a finite number of terms.

Even though operations like vector addition and scalar multiplication on the subset are typically the same as the operations on the larger parent spaces, we still often wish to work in the smaller more relevant subset rather than thinking about the larger ambient space. When does the subset behave like a vector space in its own right? In general, when is a subset of a vector space also a vector space?

## 数学代写|线性代数代写linear algebra代考|Solution Spaces

V=\left{\left(\begin{array}{l} x_{1} \ x_{2} \ x_{3} \end{array}\right) \in \mathbb{R}^{3} \中间 x_{1}+3 x_{2}-x_{3}=0\right}V=\left{\left(\begin{array}{l} x_{1} \ x_{2} \ x_{3} \end{array}\right) \in \mathbb{R}^{3} \中间 x_{1}+3 x_{2}-x_{3}=0\right}

(在1+在1)+3(在2+在2)−(在3+在3)=(在1+3在2−在3)+(在1+3在2−在3) =0+0=0

## 数学代写|线性代数代写linear algebra代考|Other Vector Spaces

\left{a_{n}\right}+\left{b_{n}\right}=\left{a_{n}+b_{n}\right} \text { and } \alpha \cdot\left{a_ {n}\right}=\left{\alpha a_{n}\right} 。\left{a_{n}\right}+\left{b_{n}\right}=\left{a_{n}+b_{n}\right} \text { and } \alpha \cdot\left{a_ {n}\right}=\left{\alpha a_{n}\right} 。

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

PetPics是一家专门从事人像摄影的宠物摄影公司，希望发布照片供客户审阅，但为了保护他们的艺术作品，他们只发布具有版权文本的电子版本。文字是

1. 齐次线性方程组的解集，其中n变量，是的子集Rn.
2. 射线照片是具有非负值的图像，表示具有给定几何形状的图像的较大矢量空间的子集。
3. 偶函数集R是函数向量空间的子集R.
4. 3 阶多项式形成向量空间的子集磷5(R).
5. 扩散焊接过程中棒上的热态（其集合是H米(R)) 形成所有可能的热状态的子集，因为温度固定在棒的末端。
6. 恰好有 10 个非零项的序列集是具有有限项数的序列集的子集。

## 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|MATHS 1011

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|Diffusion Welding and Heat States

In this section, we begin a deeper look into the mathematics for the diffusion welding application discussed in Chapter 1. Recall that diffusion welding can be used to adjoin several smaller rods into a single longer rod, leaving the final rod just after welding with varying temperature along the rod but with the ends having the same temperature. Recall that we measure the temperature along the rod and obtain a heat signature like the one seen in Figure $1.4$ of Chapter 1. Recall also, that the heat signature shows the temperature difference from the temperature at the ends of the rod. Thus, the initial signature (along with any subsequent signature) will show values of 0 at the ends.

The heat signature along the rod can be described by a function $f:[0, L] \rightarrow \mathbb{R}$, where $L$ is the length of the rod and $f(0)=f(L)=0$. The quantity $f(x)$ is the temperature difference on the rod at a position $x$ in the interval $[0, L]$. Because we are detecting and storing heat measurements along the rod, we are only able to collect finitely many such measurements. Thus, we discretize the heat signature $f$ by sampling at only $m$ locations along the bar. If we space the $m$ sampling locations equally, then for $\Delta x=\frac{L}{m+1}$, we can choose the sampling locations to be $\Delta x, 2 \Delta x, \ldots, m \Delta x$. Since the heat measurement is zero (and fixed) at the endpoints we do not need to sample there. The set of discrete heat measurements at a given time is called a heat state, as opposed to a heat signature, which, as discussed earlier, is defined at every point along the rod. We can record the heat state as the vector
$$u=\left[u_{0}, u_{1}, u_{2}, \ldots, u_{m}, u_{m+1}\right]=[0, f(\Delta x), f(2 \Delta x), \ldots, f(m \Delta x), 0]$$
Here, if $u_{j}=f(x)$ for some $x \in[0, L]$ then $u_{j+1}=f(x+\Delta x)$ and $u_{j-1}=f(x-\Delta x)$. Figure $2.15$ shows a (continuous) heat signature as a solid blue curve and the corresponding measured (discretized) heat state indicated by the regularly sampled points marked as circles.

## 数学代写|线性代数代写linear algebra代考|Function Spaces

We have seen that the set of discretized heat states of the preceding example forms a vector space. These discretized heat states can be viewed as real-valued functions on the set of $m+2$ points that are the sampling locations along the rod. In fact, function spaces such as $H_{m}(\mathbb{R})$ are very common and useful constructs for solving many physical problems. The following are some such function spaces.
Example 2.4.1 Let $\mathcal{F}={f: \mathbb{R} \rightarrow \mathbb{R}}$, the set of all functions whose domain is $\mathbb{R}$ and whose range consists of only real numbers.

We define addition and scalar multiplication (on functions) pointwise. That is, given two functions $f$ and $g$ and a real scalar $\alpha$, we define the sum $f+g$ by $(f+g)(x):=f(x)+g(x)$ and the scalar product $\alpha f$ by $(\alpha f)(x):=\alpha \cdot(f(x)) .(\mathcal{F},+, \cdot)$ is a vector space with scalars taken from $\mathbb{R}$.
Proof. Let $f, g, h \in \mathcal{F}$ and $\alpha, \beta \in \mathbb{R}$. We verify the 10 properties of Definition $2.3 .5$.

• Since $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ and based on the definition of addition, $f+g: \mathbb{R} \rightarrow \mathbb{R}$. So $f+g \in \mathcal{F}$ and $\mathcal{F}$ is closed over addition.
• Similarly, $\mathcal{F}$ is closed under scalar multiplication.
\begin{aligned} (f+g)(x) &=f(x)+g(x) \ &=g(x)+f(x) \ &=(g+f)(x) \end{aligned}
So, $f+g=g+f$
\begin{aligned} ((f+g)+h)(x) &=(f+g)(x)+h(x) \ &=(f(x)+g(x))+h(x) \ &=f(x)+(g(x)+h(x)) \ &=f(x)+(g+h)(x) \ &=(f+(g+h))(x) \end{aligned}
So $(f+g)+h=f+(g+h)$.
• We see, also, that scalar multiplication is associative. Indeed,
$$(\alpha \cdot(\beta \cdot f))(x)=(\alpha \cdot(\beta \cdot f(x)))=(\alpha \beta) f(x)=((\alpha \beta) \cdot f)(x)$$
So $\alpha \cdot(\beta \cdot f)=(\alpha \beta) \cdot f$

## 数学代写|线性代数代写linear algebra代考|Matrix Spaces

A matrix is an array of real numbers arranged in a rectangular grid, for example, let
$$A=\left(\begin{array}{lll} 1 & 2 & 3 \ 5 & 7 & 9 \end{array}\right)$$
The matrix $A$ has 2 rows (horizontal) and 3 columns (vertical), so we say it is a $2 \times 3$ matrix. In general, a matrix $B$ with $m$ rows and $n$ columns is called an $m \times n$ matrix. We say the dimensions of the matrix are $m$ and $n$.

Any two matrices with the same dimensions are added together by adding their entries entry-wise. A matrix is multiplied by a scalar by multiplying all of its entries by that scalar (that is, multiplication of a matrix by a scalar is also an entry-wise operation, as in Example 2.3.8).
Example 2.4.6 Let
$$A=\left(\begin{array}{lll} 1 & 2 & 3 \ 5 & 7 & 9 \end{array}\right), B=\left(\begin{array}{ccc} 1 & 0 & 1 \ -2 & 1 & 0 \end{array}\right), \text { and } C=\left(\begin{array}{ll} 1 & 2 \ 3 & 5 \end{array}\right)$$
Then
$$A+B=\left(\begin{array}{lll} 2 & 2 & 4 \ 3 & 8 & 9 \end{array}\right) \text {, }$$
but since $A \in \mathcal{M}{2 \times 3}$ and $C \in \mathcal{M}{2 \times 2}$, the definition of matrix addition does not work to compute $A+C$. That is, $A+C$ is undefined.
Using the definition of scalar multiplication, we get
$$3 \cdot A=\left(\begin{array}{lll} 3(1) & 3(2) & 3(3) \ 3(5) & 3(7) & 3(9) \end{array}\right)=\left(\begin{array}{ccc} 3 & 6 & 9 \ 15 & 21 & 27 \end{array}\right)$$
With this understanding of operations on matrices, we can now discuss $\left(\mathcal{M}_{m \times n},+, \cdot\right)$ as a vector space over $\mathbb{R}$.

## 数学代写|线性代数代写linear algebra代考|Function Spaces

• 自从F:R→R和G:R→R并基于加法的定义，F+G:R→R. 所以F+G∈F和F加法结束。
• 相似地，F在标量乘法下是闭合的。
• 加法是可交换的，因为
(F+G)(X)=F(X)+G(X) =G(X)+F(X) =(G+F)(X)
所以，F+G=G+F
• 并且，加法关联因为
((F+G)+H)(X)=(F+G)(X)+H(X) =(F(X)+G(X))+H(X) =F(X)+(G(X)+H(X)) =F(X)+(G+H)(X) =(F+(G+H))(X)
所以(F+G)+H=F+(G+H).
• 我们还看到，标量乘法是关联的。的确，
(一个⋅(b⋅F))(X)=(一个⋅(b⋅F(X)))=(一个b)F(X)=((一个b)⋅F)(X)
所以一个⋅(b⋅F)=(一个b)⋅F

## 数学代写|线性代数代写linear algebra代考|Matrix Spaces

3⋅一个=(3(1)3(2)3(3) 3(5)3(7)3(9))=(369 152127)

## 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|MAST10007

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 Geometry of Systems of Equations

It turns out that there is an intimate connection between solutions to systems of equations in two variables and the geometry of lines in $\mathbb{R}^{2}$. We recall the graphical method to solving systems below. Although you will likely have already done this in previous classes, we include it here so that you can put this knowledge into the context of solution sets to systems of equations as classified in Theorem 2.2.20.
We begin with the following simple example:
Example 2.2.27 Let us consider $u=\left(\begin{array}{c}2 \ -3\end{array}\right), v=\left(\begin{array}{l}1 \ 1\end{array}\right)$, and $w=\left(\begin{array}{l}2 \ 3\end{array}\right) \in \mathbb{R}^{2}$. Suppose we want to know if we can express $u$ using arithmetic operations on $v$ and $w$. In other words, we want to know if there are scalars $x, y$ so that
$$\left(\begin{array}{c} 2 \ -3 \end{array}\right)=x \cdot\left(\begin{array}{l} 1 \ 1 \end{array}\right)+y \cdot\left(\begin{array}{l} 2 \ 3 \end{array}\right) \text {. }$$
We can rewrite the right-hand side of the vector equation so that we have the equation with two vectors
$$\left(\begin{array}{c} 2 \ -3 \end{array}\right)=\left(\begin{array}{l} x+2 y \ x+3 y \end{array}\right) .$$
The equivalent system of linear equations with 2 equations and 2 variables is
\begin{aligned} &x+2 y=2 \ &x+3 y=-3 \end{aligned}
Equations (2.18) and (2.19) are equations of lines in $\mathbb{R}^{2}$, that is, the set of pairs $(x, y)$ that satisfy each equation is the set of points on each respective line. Hence, finding $x$ and $y$ that satisfy both equations amounts to finding all points $(x, y)$ that are on both lines. If we graph these two lines, we can see that they appear to cross at one point, $(12,-5)$, and nowhere else, so we estimate $x=12$ and $y=-5$ is the only solution of the two equations. (See Figure 2.9.) You can also algebraically verify that $(12,5)$ is a solution to the system.

## 数学代写|线性代数代写linear algebra代考|Images and Image Arithmetic

In Section $2.1$ we saw that if you add two images, you get a new image, and that if you multiply an image by a scalar, you get a new image. We represented a rectangular pixelated image as an array of values, or equivalently, as a rectangular array of grayscale patches. This is a very natural idea in the context of digital photography.

Recall the definition of an image given in Section 2.1. We repeat it here, and follow the definition by some examples of images with different geometric arrangements.

An image is a finite ordered list of real values with an associated geometric arrangement.
Four examples of arrays along with an index system specifying the order of patches can be seen in Figure 2.11. As an image, each patch would also have a numerical value indicating the brightness of the patch (not shown in the figure). The first is a regular pixel array commonly used for digital photography. The second is a hexagonal pattern which also nicely tiles a plane. The third is a map of the African continent and Madagascar subdivided by country. The fourth is a square pixel set with enhanced resolution toward the center of the field of interest. It should be clear from the definition that images are not matrices. Only the first example might be confused with a matrix.

We first fix a particular geometric arrangement of pixels (and let $n$ denote the number of pixels in the arrangement). Then an image is precisely described by its (ordered) intensity values. With this determined, we formalize the notions of scalar multiplication and addition on images that were developed in the previous section.

## 数学代写|线性代数代写linear algebra代考|Vectors and Vector Spaces

In the last section, we saw that the set of $4 \times 4$ images, together with real scalars, satisfies several natural properties. There are in fact many other sets of objects that also have these properties.

One class of objects with these properties are the vectors that you may have seen in a course in multivariable calculus or physics. In those courses, vectors are objects with a fixed number, say $n$, of values put together into an ordered tuple. That is, the word vector may bring to mind something that looks like $\langle a, b\rangle,\langle a, b, c\rangle$, or $\left\langle a_{1}, a_{2}, \ldots, a_{n}\right\rangle$. Maybe you have even seen notation like any of the following:
$$(a, b), \quad(a, b, c), \quad\left(a_{1}, a_{2}, \ldots, a_{n}\right),\left(\begin{array}{l} a \ b \ c \end{array}\right), \quad\left[\begin{array}{c} a \ b \ c \end{array}\right],\left(\begin{array}{c} a_{1} \ a_{2} \ \vdots \ a_{n} \end{array}\right),\left[\begin{array}{c} a_{1} \ a_{2} \ \vdots \ a_{n} \end{array}\right]$$
called vectors as well.
In this section, we generalize the notion of a vector. In particular, we will understand that images and other classes of objects can be vectors in an appropriate context. When we consider objects like brain images, radiographs, or heat state signatures, it is often useful to understand them as collections having certain natural mathematical properties. Indeed, we will develop mathematical tools that can be used on all such sets, and these tools will be instrumental in accomplishing our application tasks.
We haven’t yet made the definition of a vector space (or even a vector) rigorous. We still have some more setup to do. In this text, we will primarily use two scalar fields ${ }^{6}: \mathbb{R}$ and $Z_{2}$. The field $Z_{2}$ is the two element (or binary) set ${0,1}$ with addition and multiplication defined modulo 2 . That is, addition defined modulo 2, means that:

$$0+0=0, \quad 0+1=1+0=1, \quad \text { and } 1+1=0$$
And, multiplication defined modulo 2 means
$$0 \cdot 0=0, \quad 0 \cdot 1=1 \cdot 0=0, \quad \text { and } 1 \cdot 1=1 .$$
We can think of the two elements as “on” and “off” and the operations as binary operations. If we add 1 , we flip the switch and if we add 0 , we do nothing. We know that $Z_{2}$ is closed under scalar multiplication and vector addition.

## 数学代写|线性代数代写linear algebra代考|The Geometry of Systems of Equations

(2 −3)=X⋅(1 1)+是⋅(2 3).

(2 −3)=(X+2是 X+3是).

X+2是=2 X+3是=−3

## 数学代写|线性代数代写linear algebra代考|Vectors and Vector Spaces

(一个,b),(一个,b,C),(一个1,一个2,…,一个n),(一个 b C),[一个 b C],(一个1 一个2 ⋮ 一个n),[一个1 一个2 ⋮ 一个n]

0+0=0,0+1=1+0=1, 和 1+1=0

0⋅0=0,0⋅1=1⋅0=0, 和 1⋅1=1.

## 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|MAST10022

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|Techniques for Solving Systems of Linear Equations

In this section, we will describe two techniques for solving systems of equations. We use these two techniques to solve systems like the one presented in the previous section that arose from a question about images.
Method of elimination
In this section, we solve the system of equations in $2.4$ using the method of elimination. You may have used this method before, but we include it here to introduce some terminology to which we will refer in later sections. We will also give a parallel method later in this section.

Two systems of equations are said to be equivalent if they have the same solution set.
The idea behind the method of elimination is that we seek to manipulate the equations in a system so that the solution is easier to obtain. Specifically, in the new system, one or more of the equations will be of the form $x_{i}=c$. Since one of the equations tells us directly the value of one of the variables, we can substitute that value into the other equations and the remaining, smaller system has the same solution (together, of course, with $x_{i}=c$ ).

Before we solve the system in $(2.4)$, we provide the list of allowed operations for solving a system of equations, using the method of elimination.
Allowed operations for solving systems of equations
(1) Multiply both sides of an equation by a nonzero number.
(2) Change one equation by adding a nonzero multiple of another equation to it.
(3) Change the order of equations.
You may find these operations familiar from your earlier experience solving systems of equations; they do not change the solution set of a system. In other words, every time we change a system using one of these operations, we obtain an equivalent system of equations.

## 数学代写|线性代数代写linear algebra代考|echelon form

A matrix is in echelon form if the following three statements are true:
-All leading entries are 1 .
-Each leading entry is to the right of the leading entry in the row above it.
-Any row of zeros is below all rows that are not all zero.
The matrix $P$, above, is not in echelon form. Two of the three criteria above are not true. First, 2 is a leading entry so the first rule is not true. Second, the leading entry in the second row is to the left of the leading entry in the first row, violating the second rule.

However, if we start with $P$, then multiply row two by $1 / 2$ and then switch the first two rows, we get
$$Q=\left(\begin{array}{cccc|c} 1 & 0 & 1 / 2 & 5 / 2 & 1 \ 0 & 1 & 2 & 4 & 0 \ 0 & 0 & 0 & 1 & 1 \end{array}\right)$$
which is in echelon form. It is not just a coincidence that we could row reduce $P$ to echelon form; in fact, every matrix can be put into echelon form. It is indeed convenient that every matrix is row equivalent to a matrix that is in echelon form, and you should be able to see that an augmented matrix in echelon form corresponds to a simpler system to solve.

However, augmented matrices in echelon form are not the simplest to solve, and moreover, echelon form is not unique. For example, both the matrices $R$ and $S$ below are also in echelon form and are row equivalent to $Q$.

$$\begin{gathered} R=\left(\begin{array}{cccc|c} 1 & -1 & -3 / 2 & -1 & 3 / 2 \ 0 & 1 & 2 & 6 & 2 \ 0 & 0 & 0 & 1 & 1 \end{array}\right), \ S=\left(\begin{array}{cccc|c} 1 & 0 & 1 / 2 & 0 & -3 / 2 \ 0 & 1 & 2 & 0 & -4 \ 0 & 0 & 0 & 1 & 1 \end{array}\right) . \end{gathered}$$
Since the matrices $P, Q, R$, and $S$ are all row equivalent, they correspond to equivalent systems. Which would you rather work with? What makes your choice nicer to solve than the others? Likely, what you are discovering is that matrices corresponding to systems that are quite easy to solve are matrices in reduced echelon form.

## 数学代写|线性代数代写linear algebra代考|Elementary Matrix

In this section, we will briefly connect matrix reduction to a set of matrix products ${ }^{5}$. This connection will prove useful later. To begin, let us define an elementary matrix. We begin with the identity matrix.

An $n \times n$ elementary matrix $E$ is a matrix that can be obtained by performing one row operation on $I_{n}$.
Let us give a couple examples of elementary matrices before we give some results.
Example $2.2 .23$ The following are $3 \times 3$ elementary matrices:
$E_{1}=\left(\begin{array}{lll}1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0\end{array}\right)$ is obtained by changing the order of rows 2 and 3 of the identity matrix.
$E_{2}=\left(\begin{array}{lll}2 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right)$ is obtained by multiplying row 1 of $I_{3}$ by 2

• $E_{3}=\left(\begin{array}{lll}1 & 0 & 0 \ 3 & 1 & 0 \ 0 & 0 & 1\end{array}\right)$ is obtained by adding 3 times row 1 to row $2 .$
Since $M=\left(\begin{array}{ccc}2 & 0 & 0 \ -3 & 1 & 0 \ 0 & 0 & 1\end{array}\right)$ cannot be obtained by performing a single row operation on $I_{3}$, so is not an elementary matrix.
Let us now see how these are related to matrix reduction. Consider the following example:
Example 2.2.24 Let $M=\left(\begin{array}{ccc}2 & 3 & 5 \ 1 & 2 & 1 \ 3 & 4 & -1\end{array}\right)$. Let us see what happens when we multiply by each of the elementary matrices in Example 2.2.23.
$$E_{1} M=\left(\begin{array}{lll} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{array}\right)\left(\begin{array}{ccc} 2 & 3 & 5 \ 1 & 2 & 1 \ 3 & 4 & -1 \end{array}\right)=\left(\begin{array}{ccc} 2 & 3 & 5 \ 3 & 4 & -1 \ 1 & 2 & 1 \end{array}\right)$$
The matrix multiplication results in $M$ altered by changing rows 2 and 3 .
$$E_{2} M=\left(\begin{array}{lll} 2 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{ccc} 2 & 3 & 5 \ 1 & 2 & 1 \ 3 & 4 & -1 \end{array}\right)=\left(\begin{array}{ccc} 4 & 6 & 10 \ 1 & 2 & 1 \ 3 & 4 & -1 \end{array}\right)$$
The matrix multiplication results in $M$ altered by multiplying row 1 by 2 .

## 数学代写|线性代数代写linear algebra代考|Techniques for Solving Systems of Linear Equations

(1) 将方程两边乘以一个非零数。
(2) 通过添加另一个方程的非零倍数来改变一个方程。
(3) 改变方程的顺序。

## 数学代写|线性代数代写linear algebra代考|echelon form

-所有前导条目均为 1 。
– 每个前导条目都位于其上方行中的前导条目的右侧。
– 任何零行低于所有非全零的行。

R=(1−1−3/2−13/2 01262 00011), 小号=(101/20−3/2 0120−4 00011).

## 数学代写|线性代数代写linear algebra代考|Elementary Matrix

• 和3=(100 310 001)通过将第 1 行与第 1 行相加 3 次获得2.
自从米=(200 −310 001)不能通过对上执行单行操作来获得我3，所以不是初等矩阵。
现在让我们看看这些与矩阵约简有何关系。考虑以下示例：
示例 2.2.24 让米=(235 121 34−1). 让我们看看当我们乘以示例 2.2.23 中的每个基本矩阵时会发生什么。
和1米=(100 001 010)(235 121 34−1)=(235 34−1 121)
矩阵乘法导致米通过更改第 2 行和第 3 行来更改。
和2米=(200 010 001)(235 121 34−1)=(4610 121 34−1)
矩阵乘法导致米通过将第 1 行乘以 2 来更改。

## 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|MATH1002

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|Exploration: Digital Images

In order to understand and solve our tomography task (Section 1.2.1), we must first understand the nature of the radiographs that comprise our data. Each radiograph is actually a digitally stored collection of numerical values. It is convenient for us when they are displayed in a pixel arrangement with colors or grayscale. This section explores the nature of pixelized images and provides exercises and questions to help us understand their place in a linear algebra context.

We begin by formalizing the concept of an image with a definition. We will then consider the most familiar examples of images in this section. In subsequent sections we will revisit this definition and explore other examples.

First, let us look at an image from a camera in grayscale. In Figure $2.3$, we see one of the authors learning to sail. When we zoom in on a small patch, we see squares of uniform color. These are the pixels in the image. Each square (or pixel) has an associated intensity or brightness. Intensities are given a corresponding numerical value for storage in computer or camera memory. Brighter pixels are assigned larger numerical values.

Consider the $4 \times 4$ grayscale image in Figure 2.4. This image corresponds to the array of numbers at right, where a black pixel corresponds to intensity 0 and increasingly lighter shades of gray correspond to increasing intensity values. A white pixel (not shown) corresponds to an intensity of $16 .$

A given image can be displayed on different scales; in Figure 2.3, a pixel value of 0 is displayed as black and 255 is displayed as white, while in Figure $2.4$ a pixel value of 0 is displayed as black and 16 is displayed as white. The display scale does not change the underlying pixel values of the image.
Also, the same object may produce different images when imaged with different recording devices, or even when imaged using the same recording device with different calibrations. For example, this is what a smart phone is doing when you touch a portion of the screen to adjust the brightness when you take a picture with it.

Our definition of an image yields a natural way to think about arithmetic operations on images such as multiplication by a scalar and adding two images. For example, suppose we start with the three images A, B, and C in Figure 2.5.

Multiplying Image A by one half results in Image 1 in Figure 2.6. Every intensity value is now half what it previously was, so all pixels have become darker gray (representing their lower intensity). Adding Image 1 to Image $\mathrm{C}$ results in Image 2 (also in Figure 2.6); so Image 2 is created by doing arithmetic on Images A and $\mathrm{C}$.

Caution: Digital images and matrices are both arrays of numbers. However, not all digital images have rectangular geometric configurations like matrices ${ }^{1}$, and even digital images with rectangular configurations are not matrices, since there are operations ${ }^{2}$ that can be performed with matrices that do not make sense for digital images.

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

For some of these exercises you will need access to OCTAVE or MATLAB software. The following exercises refer to images found in Figures $2.5$ and 2.6.

1. Express Image 3 using arithmetic operations on Images A, B, and $C$. (Note that the pixel intensities in Image 3 are all either 4,8 , or 16.)
2. Express Image 4 using arithmetic operations on Images A, B, and C. (Note that the pixel intensities in Image 4 are all either 0 or 16.)
3. Input the following lines of code into the command window of OCTAVE/MATLAB. Note that ending a line with a semicolon suppresses terminal output. If you want to show the result of a computation, delete the semicolon at the end of its line. Briefly describe what each of these lines of code produces.
1. Enter the following lines of code one at a time and state what each does.
2. Write your own lines of code to check your conjectures for producing Images 3 and/or 4 . How close are these to Images 3 and/or 4?
3. We often consider display scales that assign pixels with value 0 to the color black. If a recording device uses such a scale then we do not expect any images it produces to contain pixels with negative values. However, in our definition of an image we do not restrict the pixel values. In this problem you will explore how OCTAVE/MATLAB displays an image with negative pixel values, and you will explore the effects of different gray scale ranges on an image.

Input the image pictured below into OCTAVE/MATLAB. Then display the image using each of the following five grayscale ranges.

## 数学代写|线性代数代写linear algebra代考|Systems of Equations

In Section $2.1$, we considered various $4 \times 4$ images (see page 11 ). We showed that Image 2 could be formed by performing image addition and scalar multiplication on Images $A, B$, and $C$. In particular,
$$(\text { Image } 2)=\left(\frac{1}{2}\right)(\text { Image } A)+(0)(\text { Image } B)+(1)(\text { Image } C)$$
We also posed the question about whether or not Images 3 and 4 can be formed using any arithmetic operations of Images $A, B$, and C. One can definitely determine, by inspection, the answer to these questions. Sometimes, however, trying to answer such questions by inspection can be a very tedious task. In this section, we introduce tools that can be used to answer such questions. In particular, we will discuss the method of elimination, used for solving systems of linear equations. We will also use matrix reduction on an augmented matrix to solve the corresponding system of equations. We will conclude the section with a key connection between the number of solutions to a system of equations and a reduced form of the augmented matrix.

Let $I_{1}$ and $I_{2}$ be images. We say that $I_{1}=I_{2}$ if each pair of corresponding pixels from $I_{1}$ and $I_{2}$ has the same intensity.

The convention in Figure 2.4, Definition 2.2.1, and Equation $2.1$ give us a means to write an equation, corresponding to the upper left pixel of Image D,
$$8=0 \alpha+4 \beta+8 \gamma$$
This equation has a very specific form: it is a linear equation. Such equations are at the heart of the study of linear algebra, so we recall the definition below.

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

1. 使用图像 A、B 和图像上的算术运算来表达图像 3C. （请注意，图像 3 中的像素强度都是 4,8 或 16。）
2. 使用图像 A、B 和 C 的算术运算来表达图像 4。（请注意，图像 4 中的像素强度都是 0 或 16。）
3. 在 OCTAVE/MATLAB 的命令窗口中输入以下代码行。请注意，以分号结束一行会抑制终端输出。如果要显示计算结果，请删除行尾的分号。简要描述每行代码产生的内容。
1. 一次输入以下几行代码，并说明每行的作用。
2. 编写您自己的代码行来检查您对生成图像 3 和/或 4 的猜想。这些与图像 3 和/或 4 有多接近？
3. 我们经常考虑将值为 0 的像素分配给黑色的显示比例。如果记录设备使用这样的比例，那么我们不希望它产生的任何图像包含具有负值的像素。但是，在我们对图像的定义中，我们不限制像素值。在这个问题中，您将探索 OCTAVE/MATLAB 如何显示具有负像素值的图像，并且您将探索不同灰度范围对图像的影响。

## 数学代写|线性代数代写linear algebra代考|Systems of Equations

( 图片 2)=(12)( 图片 一个)+(0)( 图片 乙)+(1)( 图片 C)

8=0一个+4b+8C

## 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。