## 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|ROW REDUCTION AND ECHELON FORMS

This section refines the method of Section $1.1$ into a row reduction algorithm that will enable us to analyze any system of linear equations. ${ }^1$ By using only the first part of the algorithm, we will be able to answer the fundamental existence and uniqueness questions posed in Section 1.1.

The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system. So the first part of this section concerns an arbitrary rectangular matrix and begins by introducing two important classes of matrices that include the “triangular” matrices of Section 1.1. In the definitions that follow, a nonzero row or column in a matrix means a row or column that contains at least one nonzero entry; a leading entry of a row refers to the leftmost nonzero entry (in a nonzero row).

An echelon matrix (respectively, reduced echelon matrix) is one that is in echelon form (respectively, reduced echelon form). Property 2 says that the leading entries form an echelon (“steplike”) pattern that moves down and to the right through the matrix. Property 3 is a simple consequence of property 2 , but we include it for emphasis.
The “triangular” matrices of Section 1.1, such as
$$\left[\begin{array}{rrrc} 2 & -3 & 2 & 1 \ 0 & 1 & -4 & 8 \ 0 & 0 & 0 & 5 / 2 \end{array}\right] \text { and }\left[\begin{array}{lllr} 1 & 0 & 0 & 29 \ 0 & 1 & 0 & 16 \ 0 & 0 & 1 & 3 \end{array}\right]$$
are in echelon form. In fact, the second matrix is in reduced echelon form. Here are additional examples.

EXAMPLE 1 The following matrices are in echelon form. The leading entries ( $\boldsymbol{)}$ ) may have any nonzero value; the starred entries $(*)$ may have any value (including zero).

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

The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system.
Suppose, for example, that the augmented matrix of a linear system has been changed into the equivalent reduced echelon form
$$\left[\begin{array}{rrrr} 1 & 0 & -5 & 1 \ 0 & 1 & 1 & 4 \ 0 & 0 & 0 & 0 \end{array}\right]$$
There are three variables because the augmented matrix has four columns. The associated system of equations is
$$\begin{array}{r} x_1-5 x_3=1 \ x_2+x_3=4 \ 0=0 \end{array}$$
The variables $x_1$ and $x_2$ corresponding to pivot columns in the matrix are called basic variables. ${ }^2$ The other variable, $x_3$, is called a free variable.

Whenever a system is consistent, as in (4), the solution set can be described explicitly by solving the reduced system of equations for the basic variables in terms of the free variables. This operation is possible because the reduced echelon form places each basic variable in one and only one equation. In (4), solve the first equation for $x_1$ and the second for $x_2$. (Ignore the third equation; it offers no restriction on the variables.)
$$\left{\begin{array}{l} x_1=1+5 x_3 \ x_2=4-x_3 \ x_3 \text { is free } \end{array}\right.$$
The statement ” $x_3$ is free” means that you are free to choose any value for $x_3$. Once that is done, the formulas in (5) determine the values for $x_1$ and $x_2$. For instance, when $x_3=0$, the solution is $(1,4,0)$; when $x_3=1$, the solution is $(6,3,1)$. Each different choice of $x_3$ determines a (different) solution of the system, and every solution of the system is determined by a choice of $x_3$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|ROW REDUCTION AND ECHELON FORMS

$1.1$ 节的“三角”矩阵，如

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

$$\left[\begin{array}{llllllllllll} 1 & 0 & -5 & 1 & 0 & 1 & 1 & 4 & 0 & 0 & 0 & 0 \end{array}\right]$$

$$x_1-5 x_3=1 x_2+x_3=40=0$$

x_1=1+5 x_3 x_2=4-x_3 x_3 \text { is free }
$$正确的。 \ \$$

## 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|Solving a Linear System

This section and the next describe an algorithm, or a systematic procedure, for solving linear systems. The basic strategy is to replace one system with an equivalent system (i.e., one with the same solution set) that is easier to solve.

Roughly speaking, use the $x_1$ term in the first equation of a system to eliminate the $x_1$ terms in the other equations. Then use the $x_2$ term in the second equation to eliminate the $x_2$ terms in the other equations, and so on, until you finally obtain a very simple equivalent system of equations.

Three basic operations are used to simplify a linear system: Replace one equation by the sum of itself and a multiple of another equation, interchange two equations, and multiply all the terms in an equation by a nonzero constant. After the first example, you will see why these three operations do not change the solution set of the system.

Row operations can be applied to any matrix, not merely to one that arises as the augmented matrix of a linear system. Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other.
It is important to note that row operations are reversible. If two rows are interchanged, they can be returned to their original positions by another interchange. If a row is scaled by a nonzero constant $c$, then multiplying the new row by $1 / c$ produces the original row. Finally, consider a replacement operation involving two rows -say, rows 1 and 2 -and suppose that $c$ times row 1 is added to row 2 to produce a new row 2. To “reverse” this operation, add $-c$ times row 1 to (new) row 2 and obtain the original row 2. See Exercises $29-32$ at the end of this section.

At the moment, we are interested in row operations on the augmented matrix of a system of linear equations. Suppose a system is changed to a new one via row operations. By considering each type of row operation, you can see that any solution of the original system remains a solution of the new system. Conversely, since the original system can be produced via row operations on the new system, each solution of the new system is also a solution of the original system. This discussion justifies the following statement.

## 数学代写|线性代数代写linear algebra代考|Existence and Uniqueness Questions

Section $1.2$ will show why a solution set for a linear system contains either no solutions, one solution, or infinitely many solutions. Answers to the following two questions will determine the nature of the solution set for a linear system.

To determine which possibility is true for a particular system, we ask two questions.

These two questions will appear throughout the text, in many different guises. This section and the next will show how to answer these questions via row operations on the augmented matrix.
EXAMPLE 2 Determine if the following system is consistent:
\begin{aligned} x_1-2 x_2+x_3 & =0 \ 2 x_2-8 x_3 & =8 \ 5 x_1-5 x_3 & =10 \end{aligned}
SOLUTION This is the system from Example 1. Suppose that we have performed the row operations necessary to obtain the triangular form
\begin{aligned} x_1-2 x_2+x_3 & =0 \ x_2-4 x_3 & =4 \ x_3 & =-1 \end{aligned} \quad\left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \ 0 & 1 & -4 & 4 \ 0 & 0 & 1 & -1 \end{array}\right] At this point, we know $x_3$. Were we to substitute the value of $x_3$ into equation 2 , we could compute $x_2$ and hence could determine $x_1$ from equation 1 . So a solution exists; the system is consistent. (In fact, $x_2$ is uniquely determined by equation 2 since $x_3$ has only one possible value, and $x_1$ is therefore uniquely determined by equation 1 . So the solution is unique.)

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Existence and Uniqueness Questions

$$x_1-2 x_2+x_3=02 x_2-8 x_3 \quad=85 x_1-5 x_3=10$$

$$x_1-2 x_2+x_3=0 x_2-4 x_3 \quad=4 x_3=-1 \quad\left[\begin{array}{llllllllllll} 1 & -2 & 1 & 0 & 0 & 1 & -4 & 4 & 0 & 0 & 1 & -1 \end{array}\right]$$

## 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|SYSTEMS OF LINEAR EQUATIONS

A linear equation in the variables $x_1, \ldots, x_n$ is an equation that can be written in the form
$$a_1 x_1+a_2 x_2+\cdots+a_n x_n=b$$
where $b$ and the coefficients $a_1, \ldots, a_n$ are real or complex numbers, usually known in advance. The subscript $n$ may be any positive integer. In textbook examples and exercises, $n$ is normally between 2 and 5 . In real-life problems, $n$ might be 50 or 5000 , or even larger.
The equations
$$4 x_1-5 x_2+2=x_1 \quad \text { and } \quad x_2=2\left(\sqrt{6}-x_1\right)+x_3$$
are both linear because they can be rearranged algebraically as in equation (1):
$$3 x_1-5 x_2=-2 \text { and } 2 x_1+x_2-x_3=2 \sqrt{6}$$
The equations
$$4 x_1-5 x_2=x_1 x_2 \quad \text { and } \quad x_2=2 \sqrt{x_1}-6$$
are not linear because of the presence of $x_1 x_2$ in the first equation and $\sqrt{x_1}$ in the second. A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables-say, $x_1, \ldots, x_n$. An example is
$$\begin{array}{r} 2 x_1-x_2+1.5 x_3=8 \ x_1-4 x_3=-7 \end{array}$$ A solution of the system is a list $\left(s_1, s_2, \ldots, s_n\right)$ of numbers that makes each equation a true statement when the values $s_1, \ldots, s_n$ are substituted for $x_1, \ldots, x_n$, respectively. For instance, $(5,6.5,3)$ is a solution of system ( 2 ) because, when these values are substituted in (2) for $x_1, x_2, x_3$, respectively, the equations simplify to $8=8$ and $-7=-7$.

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

The essential information of a linear system can be recorded compactly in a rectangular array called a matrix. Given the system
\begin{aligned} x_1-2 x_2+x_3 & =0 \ 2 x_2-8 x_3 & =8 \ 5 x_1-5 x_3 & =10 \end{aligned}
with the coefficients of each variable aligned in columns, the matrix
$$\left[\begin{array}{rrr} 1 & -2 & 1 \ 0 & 2 & -8 \ 5 & 0 & -5 \end{array}\right]$$
is called the coefficient matrix (or matrix of coefficients) of the system (3), and
$$\left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \ 0 & 2 & -8 & 8 \ 5 & 0 & -5 & 10 \end{array}\right]$$
is called the augmented matrix of the system. (The second row here contains a zero because the second equation could be written as $0 \cdot x_1+2 x_2-8 x_3=8$.) An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations.

The size of a matrix tells how many rows and columns it has. The augmented matrix (4) above has 3 rows and 4 columns and is called a $3 \times 4$ (read “3 by 4 “) matrix. If $m$ and $n$ are positive integers, an $\boldsymbol{m} \times \boldsymbol{n}$ matrix is a rectangular array of numbers with $m$ rows and $n$ columns. (The number of rows always comes first.) Matrix notation will simplify the calculations in the examples that follow.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|SYSTEMS OF LINEAR EQUATIONS

$$a_1 x_1+a_2 x_2+\cdots+a_n x_n=b$$

$$4 x_1-5 x_2+2=x_1 \quad \text { and } \quad x_2=2\left(\sqrt{6}-x_1\right)+x_3$$

$$3 x_1-5 x_2=-2 \text { and } 2 x_1+x_2-x_3=2 \sqrt{6}$$

$$4 x_1-5 x_2=x_1 x_2 \quad \text { and } \quad x_2=2 \sqrt{x_1}-6$$

$$2 x_1-x_2+1.5 x_3=8 x_1-4 x_3=-7$$

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

$$x_1-2 x_2+x_3=02 x_2-8 x_3=85 x_1-5 x_3=10$$

$$\left[\begin{array}{llllllll} 1 & -2 & 1 & 0 & 2 & -85 & 0 & -5 \end{array}\right]$$

$$\left[\begin{array}{lllllllllll} 1 & -2 & 1 & 0 & 0 & 2 & -8 & 85 & 0 & -5 & 10 \end{array}\right]$$

## 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|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|MATRIX MULTIPLICATION

Here, we present another operation applicable in $M_{m n}$ in which the inputs are two matrices and the output is another matrix. Although this is not an operation indicative of a vector space, it is an essential ingredient in what will follow.

Definition $1.11$ Let $A=\left[a_{i j}\right] \in M_{m n}$ and $B=\left[b_{i j}\right] \in M_{n r}$. Then the product $C=\left[c_{i j}\right]=A B \in M_{m r}$ is defined as follows:
$$c_{i j}=\sum_{k=1}^n a_{i k} b_{k j} .$$
Notice that to perform matrix multiplication on matrices, it is necessary that the number of columns in $A$ be equal to the number of rows in $B$ and the resulting matrix has the same number of rows as $A$ and the same number of columns as $B$. Perhaps a simpler way to remember the entries of $C$ is that the ijth entry of $C$ is obtained by taking the dot product of the $i$ th row of $A$ with the $j$ th column of $B$. Conversely, one can define dot product in terms of matrix multiplication. Indeed, if $v, w \in \mathbb{R}^n$, then $v \cdot w=v^T w$, where $v$ and $w$ are viewed as $n \times 1$ column matrices. This is sometimes a useful representation of dot product when demonstrating certain proofs.
Example $1.10$
$$\left[\begin{array}{lll} 1 & 2 & 3 \ 4 & 5 & 6 \end{array}\right]\left[\begin{array}{rrr} 1 & -1 & 1 \ -1 & 0 & 1 \ 0 & 1 & 1 \end{array}\right]$$
$$=\left[\begin{array}{lll} (1)(1)+(2)(-1)+(3)(0) & (1)(-1)+(2)(0)+(3)(1) & (1)(1)+(2)(1)+(3)(1) \ (4)(1)+(5)(-1)+(6)(0) & (4)(-1)+(5)(0)+(6)(1) & (4)(1)+(5)(1)+(6)(1) \end{array}\right]$$
$$=\left[\begin{array}{rrr} -1 & 2 & 6 \ -1 & 2 & 15 \end{array}\right]$$

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

We are ready to present a systematic way for solving systems of linear equations. This method is simple and will be used quite regularly throughout the remainder of the book. First, recall that every system of linear equations has an associated augmented matrix:
Example 2.2 The augmented matrix associated with the linear system
$$\left{\begin{array}{rlr} 2 x_1+x_2-x_3 & =0 \ x_1-3 x_2+x_3 & =7 \ -3 x_1+x_2+x_3 & = & -5 \end{array}\right.$$
is
$$\left[\begin{array}{rrr|r} 2 & 1 & -1 & 0 \ 1 & -3 & 1 & 7 \ -3 & 1 & 1 & -5 \end{array}\right]$$
In solving a linear system we wish to manipulate the equations without altering the solution set and arrive at a more “desirable” system of equations for which we can readily identify the solution set. The operations below achieve this goal.

Definition 2.3 The following three operations are called elementary row operations which can be applied to a system of linear equations or the associated augmented matrix:

1. Multiplying the ith equation (or ith row of the augmented matrix) by a non-zero scalar $a$. The notation is a$R_i$.
2. Switching the $i$ th and $j$ th equation (or ith and $j$ th row of the augmented matrix). The notation is $R_i \leftrightarrow R_j$.
3. Adding a scalar a times the ith equation to the $j$ th equation (or adding a times the ith row to the $j$ th row of the augmented matrix). The notation is a $R_i+R_j$.

# 线性代数代考

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

$$c_{i j}=\sum_{k=1}^n a_{i k} b_{k j} .$$

$v \cdot w=v^T w$ ，在哪里 $v$ 和 $w$ 被视为 $n \times 1$ 列矩阵。在演示某些证明时，这有时是点积的有用 表示。

\begin{aligned} & =[(1)(1)+(2)(-1)+(3)(0) \quad(1)(-1)+(2)(0)+(3)(1) \quad(1)(1)+(2)(1)+(3) \ & \end{aligned}

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

$\$ \$$Veft {$$
2 x_1+x_2-x_3=0 x_1-3 x_2+x_3=7-3 x_1+x_2+x_3=-5
$$、正确的。 is 剩下[$$
\begin{array}{lll|l|l|l|ll|l|l|l}
2 & 1 & -1 & 0 & 1 & -3 & 1 & 7-3 & 1 & 1 & -5
\end{array}
$$Iright] \ \$$

1. 将第 $\mathrm{i}$ 个方程 (或增广矩阵的第 $\mathrm{i}$ 行) 乘以非零标量 $a$. 该符号是 $R_i$.
2. 切换 $i$ 和 $j$ 第方程 (或第 $\mathrm{i}$ 和 $j$ 增广矩阵的第 th 行) 。符号是 $R_i \leftrightarrow R_j$.
3. 添加一个标量 $a$ 乘以第 $\mathrm{i}$ 个方程到 $j$ th 等式 (或将第 $\mathrm{i}$ 行的 $a$ 乘以 $j$ 增广矩阵的第 th 行)。该符号是 $R_i+R_j$.

## 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|APPLICATION: GEOMETRY

As we have already stated tuples in $\mathbb{R}^n$ along with their operations take on a geometric meaning. This section is devoted to further exploration of this observation. Recall briefly the following geometric facts about tuples:

1. A vector, $u$, can be viewed physically as an arrow.
2. The sum and difference of two vectors, $u+v$ and $u-v$, comprise the diagonals of a parallelogram whose adjacent sides are these two vectors.
3. The magnitude of a vector, $|u|$, corresponds to the length of the arrow representing $u$.
4. For vectors $u$ and $v$, we have the equation $u \cdot v=|u||v| \cos \theta$, where $\theta$ is the smaller angle between $u$ and $v$.
5. Two vectors $u$ and $v$ are parallel iff $u=a v$ or $v=a u$ for some real number $a$.
6. Two vectors $u$ and $v$ are perpendicular iff $u \cdot v=0$.
7. The vector $-u$ points in the opposite direction of $u$.
Only in this section will we allow vectors which do not have their initial point at the origin so that we can derive some nice geometric results. In this case, we will say that two vectors are equal if they have the same length and are point in the same direction.

For instance, in Figure 1.5 we have depicted a collection of vectors which are all equal to each other.

We need to introduce some notation. If $A$ and $B$ are points in space, then $\overrightarrow{A B}$ denotes the vector with initial point $A$ and terminal point $B$ as depicted in Figure 1.6.
From our discussion of the parallelogram earlier, it is clear that if $u-$ $\left[a_1, a_2, \ldots, a_n\right]$ is a vector with terminal point at $A$ and $v=\left[b_1, b_2, \ldots, b_n\right]$ is a vector with terminal point at $B$, then
$$\overrightarrow{A B}=v-u=\left[b_1-a_1, b_2-a_2, \ldots, b_n-a_n\right] .$$
With just these few facts we are capable of proving many standard geometric results.

## 数学代写|线性代数代写linear algebra代考|SECOND VECTOR SPACE: MATRICES

Here now is our second example of what later will be called a vector space. First we define a matrix.

Definition $1.8$ An $m \times n$ matrix is a rectangular array of scalars consisting of $m$ rows and $n$ columns. We say the dimensions of the matrix are ” $m$-by- $n$ or $m \times n$. .”
Example $1.8\left[\begin{array}{rrr}-1 & \pi & 6 \ \sqrt{3} & -1.2 & 3 / 4\end{array}\right]$ is an example of a $2 \times 3$ matrix.
There are several useful ways of representing a matrix. The most descriptive (and most cumbersome) is the following:
$$\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \vdots & \vdots & \ddots & \vdots \ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right]$$
Each scalar $a_{i j}$ is called the $i j$ th entry of the matrix where $1 \leq i \leq m$ and $1 \leq j \leq n$. A simpler notation for a matrix is $\left[a_{i j}\right]$. We often represent a matrix simply by $A$. Another useful way to represent a matrix is by its rows or by its columns:
$$A=\left[\begin{array}{c} r_1 \ r_2 \ \vdots \ r_m \end{array}\right], \text { where } r_i=\left[\begin{array}{llll} a_{i 1} & a_{i 2} & \cdots & a_{i n} \end{array}\right] \quad(i=1,2, \ldots, m), \text { or }$$

$$A=\left[\begin{array}{llll} c_1 & c_2 & \cdots & c_n \end{array}\right] \text {, where } c_j=\left[\begin{array}{c} a_{1 j} \ a_{2 j} \ \vdots \ a_{m j} \end{array}\right] \quad(j=1,2, \ldots, n) .$$
We are now ready to define our second vector space.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|APPLICATION: GEOMETRY

1. 一个向量, $u$, 在物理上可以看作是一个箭头。
2. 两个向量的和与差， $u+v$ 和 $u-v$ ，包括平行四边形的对角线，其相邻边是这两个向 量。
3. 矢量的大小， $|u|$, 对应于代表箭头的长度 $u$.
4. 对于载体 $u$ 和 $v$ ，我们有方程 $u \cdot v=|u||v| \cos \theta$ ，在哪里 $\theta$ 是之间的较小角度 $u$ 和 $v$.
5. 两个向量 $u$ 和 $v$ 是平行的当且仅当 $u=a v$ 要么 $v=a u$ 对于一些实数 $a$.
6. 两个向量 $u$ 和 $v$ 是垂直的当且仅当 $u \cdot v=0$.
7. 载体 $-u$ 指向相反的方向 $u$.
仅在本节中，我们将允许初始点不在原点的向量，以便我们可以得出一些不错的几何结 果。在这种情况下，如果两个向量具有相同的长度并且指向相同的方向，我们就说它们 相等。
例如，在图 $1.5$ 中，我们描绘了一组彼此相等的向量。
我们需要引入一些符号。如果 $A$ 和 $B$ 是空间中的点，那么 $\overrightarrow{A B}$ 表示具有初始点的向量 $A$ 和终点 $B$ 如图 1.6 所示。
从我们之前对平行四边形的讨论中可以清楚地看出，如果 $u-\left[a_1, a_2, \ldots, a_n\right]$ 是一个向量， 其终点位于 $A$ 和 $v=\left[b_1, b_2, \ldots, b_n\right]$ 是一个向量，其终点位于 $B$ ，然后
$$\overrightarrow{A B}=v-u=\left[b_1-a_1, b_2-a_2, \ldots, b_n-a_n\right] .$$
仅凭这几个事实，我们就能够证明许多标准的几何结果。

## 数学代写|线性代数代写linear algebra代考|SECOND VECTOR SPACE: MATRICES

$A=\left[\begin{array}{llll}r_1 r_2 & \vdots & r_m\end{array}\right]$, where $r_i=\left[\begin{array}{llll}a_{i 1} & a_{i 2} & \cdots & a_{i n}\end{array}\right] \quad(i=1,2, \ldots, m)$, or
$$A=\left[\begin{array}{llll} c_1 & c_2 & \cdots & c_n \end{array}\right], \text { where } c_j=\left[\begin{array}{c} a_{1 j} a_{2 j} \vdots a_{m j} \end{array}\right] \quad(j=1,2, \ldots, n) .$$

## 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|FIRST VECTOR SPACE: TUPLES

Here now is our first example of what later will be called a vector space. A notion in linear algebra of some importance is the scalar. For most of our discussion, a scalar will just be a real number and, at times, a complex number. A more comprehensive and perhaps advanced treatise on linear algebra would assume a scalar to be a element of what is called a field. Roughly speaking, a field gathers together some of the essential properties (or axioms) of the real numbers. We list these properties below:
Definition $1.1$ A field is a set of objects $F$ together with two operations $+$ and . (called addition and multiplication) having the following properties:
Closure: For all $a, b \in F$, we have $a+b \in F$ and $a \cdot b \in F$.
Commutativity: For all $a, b \in F$, we have $a+b=b+a$ and $a \cdot b=b \cdot a$.
Associativity: For all $a, b, c \in F$, we have $a+(b+c)=(a+b)+c$ and $a \cdot(b \cdot c)=(a \cdot b) \cdot c$.

Identity: There exist $0,1 \in F$ such that for all $a \in F$, we have $a+0-a$ and $a \cdot 1=a$.

Inverse: For every $a \in F$ there exists $b \in F$ such that $a+b=0$ ( $b$ is called the additive inverse of a) and for every $0 \neq a \in F$ there exists $b \in F$ such that $a \cdot b=1$ ( $b$ is called the multiplicative inverse of $a$ ).
Distribution: For all $a, b, c \in F$, we have $a \cdot(b+c)=a \cdot b+a \cdot c$.
The main examples of fields addressed in this text are the real numbers and the complex numbers (one can easily check that the properties above are satisfied in each example). At times we may want to prove results in more generality without assuming what field we have, but as stated, a scalar for the time being is simply another name for a real number. The standard notation for real numbers is $\mathbb{R}$.

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

Here we present another operation applicable in $\mathbb{R}^n$ in which the inputs are two vectors and the output is a scalar. The various names of this operation are dot, scalar or inner product. Although this is not an operation indicative of a vector space, it is an essential ingredient of what we will later call an inner product space.

Definition 1.4 Let $u=\left[a_1, \ldots, a_n\right], v=\left[b_1, \ldots, b_n\right] \in \mathbb{R}^n$. The dot product of $u$ and $v$, written
$$u \cdot v=a_1 b_1+\cdots+a_n b_n .$$
Example $1.3$ In $\mathbb{R}^4$,
$$\begin{gathered} {[2,25,-1,-1.3] \cdot[-3,1 / 5,3,10]=(2)(-3)+(25)(1 / 5)+(-1)(3)+(-1.3)(10)} \ =-6+5-3-13=-17 . \end{gathered}$$
The following result summarizes some elementary properties of the dot product:
Theorem 1.2 If $u, v, w \in \mathbb{R}^n$ and $a \in \mathbb{R}$, then
i. $u \cdot v=v \cdot u$.
ii. $u \cdot(v+w)=u \cdot v+u \cdot w$.
iii. $a(u \cdot v)=(a u) \cdot v=u \cdot(a v)$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|FIRST VECTOR SPACE: TUPLES

$a \cdot(b \cdot c)=(a \cdot b) \cdot c$.

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

$$u \cdot v=a_1 b_1+\cdots+a_n b_n .$$

$$[2,25,-1,-1.3] \cdot[-3,1 / 5,3,10]=(2)(-3)+(25)(1 / 5)+(-1)(3)+(-1.3)(10)$$

\begin{aligned} & \text { 二. } u \cdot(v+w)=u \cdot v+u \cdot w \ & \text { 三. } a(u \cdot v)=(a u) \cdot v=u \cdot(a v) \end{aligned}

## 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 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
$$\left(\begin{array}{c} 2 \ -3 \end{array}\right)=x \cdot\left(\begin{array}{l} 1 \ 1 \end{array}\right)+y \cdot\left(\begin{array}{l} 2 \ 3 \end{array}\right) .$$
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.

Given two images $x$ and $y$ with (ordered) intensity values $\left(x_1, x_2, \cdots, x_n\right)$ and $\left(y_1, y_2, \cdots, y_n\right)$, respectively, and the same geometry, the image sum written $z=x+y$ is the image with intensity values $z_i=x_i+y_i$ for all $i \in{1,2, \cdots, n}$, and the same geometry.

Hence, the sum of two images is the image that results from pixel-wise addition of intensity values. Put another way, the sum of two images is the image that results from adding corresponding values of their ordered lists, while maintaining the geometric arrangement of pixels.

# 线性代数代考

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

$$\left(\begin{array}{c}) 2 \ -3 \end{array}right）=x\cdot\left(begin{array}{l}) 1 \ 1 \end{array}right）+y cdot\left(begin{array}{l}) 2 \ 3 \end{array}right）。$$

$$\left(\begin{array}{c}) 2 \ -3 \end{array}right）=left(begin{array}{l}) x+2 y x+3 y \end{array}right）。$$

\begin{aligned} & x+2 y=2 & x+3 y=-3 。 & x+3 y=-3 。 \end{aligned}

## 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|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|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.
Definition 2.2.21
The $n \times n$ identity matrix, $I_n$ is the matrix that satisfies $I_n M=M I_n=M$ for all $n \times n$ matrices $M$.

One can show, by inspection, that $I_n$ must be the matrix with $n$ 1’s down the diagonal and 0 ‘s everywhere else:
$$I=\left(\begin{array}{cccc} 1 & 0 & \ldots & 0 \ 0 & 1 & & 0 \ \vdots & \ddots & \vdots \ 0 & \ldots & 1 \end{array}\right)$$

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.

# 线性代数代考

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

(1) 将方程的两边乘以一个非零数。
(2) 通过向其中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|Elementary Matrix

2.2.21
$n \times n$ 单位矩阵， $I_n$ 是满足的矩阵 $I_n M=M I_n=M$ 对所有人 $n \times n$ 矩阵 $M$.

$$I=\left(\begin{array}{ccccccccccc} 1 & 0 & \ldots & 0 & 0 & 1 & 0 & \ddots & \vdots 0 & \ldots & 1 \end{array}\right)$$

• $E_3=\left(\begin{array}{lllllllll}1 & 0 & 0 & 3 & 1 & 0 & 0 & 0 & 1\end{array}\right)$ 通过将第 1 行添加到第 2 行获得 3 次。
自从 $M=\left(\begin{array}{lllllllll}2 & 0 & 0 & -3 & 1 & 0 & 0 & 0 & 1\end{array}\right)$ 不能通过执行单行操作获得 $I_3$ ，所以不是初等矩 阵。
现在让我们看看这些与矩阵约简的关系。考虑以下示例: 每个初等矩阵时会发生什么。

## 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|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 $\mathrm{C}$ 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 $C$.

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|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.

In this section we return to one of the tasks from Section 2.1. In that task, we were asked to determine whether a particular image could be expressed using arithmetic operations on Images A, B, and C. Let us consider a similar question. Suppose we are given the images in Figures $2.7$ and 2.8. Can Image C be expressed using arithmetic operations on Images A, D, and F?
For this question, we are asking whether we can find real numbers $\alpha, \beta$, and $\gamma$ so that First, in order to make sense of this question, we need to define what it means for images to be equal.

# 线性代数代考

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

$$(\text { Image } 2)=\left(\frac{1}{2}\right)(\text { Image A })+(0)(\text { Image B })+(1)(\text { Image C }) \text {. }$$

## 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

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

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

Polynomials play several different roles in linear algebra. They define elements in $\mathcal{F}(\mathbb{R})$, for example, such as $f(x)=x^2$. Strictly speaking, this is not a polynomial but a polynomial function. This may seem like a distinction without a difference but polynomial functions over arbitrary fields can be wildly different from their underlying polynomials. Tools we have developed to this point will facilitate our introduction to polynomials.
Fix a field $\mathbb{F}$ and symbols $t, t^2, t^3, \ldots$. Let
$$\mathbb{F}[t]:=\left{c_0+c_1 t+\cdots+c_k t^k \mid c_i \in \mathbb{F}\right} .$$
Elements in $\mathbb{F}[t]$ are polynomials. Since we do not actually calculate a sum when presented with a polynomial, we say that polynomials are formal sums. We call $t$ an indeterminate or variable. Defining $t^0:=1$ lets us write
$$c_0+c_1 t+\cdots+c_k t^k=\sum_{i=0}^k c_i t^i .$$
Define addition in $\mathbb{F}[t]$ by adding coefficients of like terms,
$$\sum_{i=0}^k c_i t^i+\sum_{i=0}^k b_i t^i:=\sum_{i=0}^k\left(c_i+b_i\right) t^i .$$
Scaling in $\mathbb{F}[t]$ is done term by term:
$$a \sum_{i=0}^k c_i t^i:=\sum_{i=0}^k a c_i t^i$$
for any $a$ in $\mathbb{F}$. The zero polynomial is the polynomial for which every coefficient is zero. Two polynomials are equal provided they have identical nonzero terms. Under these definitions, $\mathbb{F}[t]$ is a countably infinite-dimensional vector space with basis $\mathcal{B}=\left{1, t, t^2, t^3, \ldots\right}$. We call $\mathbb{F}[t]$ the space of polynomials in $t$ over $\mathbb{F}$.
The $n$th term of $p(t)=\sum_{i=1}^k c_i t^i$ in $\mathbb{F}[t]$ is the monomial $c_n t^n$. The $n$th coefficient of $p(t)$ is $c_n$. A nonzero term is one with a nonzero coefficient. The constant term of $p(t)$ is $c_0$. A constant polynomial has the form $p(t)=c_0$.
The degree of $p(t)=\sum_{i=1}^k c_i t^i \neq 0$ is the maximum value of $k$ in $\mathbb{Z}_{\geq 0}$ such that $c_k \neq 0$. In this case, we write $\operatorname{deg} p(t)=k$. The degree of the zero polynomial is defined to be $-\infty$.

## 数学代写|线性代数代写linear algebra代考|R and C in Linear Algebra

Arbitrary fields are interesting and important in applications inside and outside of mathematics. Partly because of their roles in physics and geometry, though, the real numbers and the complex numbers have a special place in linear algebra. We use $\mathbb{R}^2$ to model the Euclidean plane and $\mathbb{R}^3$ to model the physics of motion at the human scale. Complex numbers underlie the theories of electricity, magnetism, and quantum mechanics, all of which use linear algebra one way or another.

We have seen that every complex vector space is a real vector space. The next theorem is more specific.
Theorem 1.49. If $V$ is a complex vector space, then
$$\operatorname{dim}{\mathbb{R}} V=2 \operatorname{dim}{\mathbb{C}} V .$$
Proof. Let $V$ be a complex vector space with basis $\mathcal{B}$. Let $i \mathcal{B}={i \mathbf{b} \mid \mathbf{b} \in \mathcal{B}}$. We claim that $\mathcal{B} \cup i \mathcal{B}$ is linearly independent over $\mathbb{R}$.
Suppose $\mathbf{b}1, \ldots, \mathbf{b}_n, \mathbf{b}{n+1}, \ldots, \mathbf{b}{n+m}$ are elements in $\mathcal{B}$ and that (1.10) $\quad a_1 \mathbf{b}_1+\cdots+a_n \mathbf{b}_n+i c_1 \mathbf{b}{n+1}+\cdots+i c_m \mathbf{b}_{n+m}=\mathbf{0}$,
for real coefficients $a_j$ and $c_j$. Since $a_j$ and $i c_j$ are also complex, we can reindex the $\mathbf{b}_j$ s if necessary to rewrite (1.10) in the form
$$z_1 \mathbf{b}_1+\cdots+z_k \mathbf{b}_k=\mathbf{0},$$
where $z_j=a_j+i c_j$ and $\mathbf{b}_1, \ldots, \mathbf{b}_k$ are distinct in $\mathcal{B}$. Since $\mathcal{B}$ is linearly independent, $z_j=0$ for all $j$, implying that all $a_j$ and $c_j$ in (1.10) are zero. This is enough to prove our claim.

A similar argument shows that any linear combination of elements in $\mathcal{B}$ over $\mathbb{C}$ can be written as a linear combination of elements in $\mathcal{B} \cup i \mathcal{B}$ over $\mathbb{R}$. This is enough to prove that $\mathcal{B} \cup i \mathcal{B}$ is a basis for $V$ over $\mathbb{R}$. The theorem follows.
Example 1.50. Let $\mathcal{B}={(1, i),(1,-1)} \subseteq \mathbb{C}^2$. Given $z_1, z_2$ in $\mathbb{C}$, we have
$$z_1(1, i)+z_2(1,-1)=\left(z_1+z_2, i z_1-z_2\right) .$$
If this linear combination is equal to the zero vector, then we can solve the following system of equations over $\mathbb{C}$ to find the coefficients:
\begin{aligned} z_1+z_2 &=0 \ i z_1-z_2 &=0 . \end{aligned}
Adding the two equations we get $(1+i) z_1=0$. Since $\mathbb{C}$ is a field, $z_1=0$, from which it follows that $z_2$ is zero. This establishes that $\mathcal{B}$ is linearly independent. Since $\mathbb{C}^2$ is 2-dimensional over $\mathbb{C}, \mathcal{B}$ must be a basis for $\mathbb{C}^2$.

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

$$c_0+c_1 t+\cdots+c_k t^k=\sum_{i=0}^k c_i t^i$$

$$\sum_{i=0}^k c_i t^i+\sum_{i=0}^k b_i t^i:=\sum_{i=0}^k\left(c_i+b_i\right) t^i$$

$$a \sum_{i=0}^k c_i t^i:=\sum_{i=0}^k a c_i t^i$$

Imathcal{B}=Vleft $\left{1, t, t^{\wedge} 2, t^{\wedge} 3\right.$, Vdots \right } } \text { . 我们称之为 } \mathbb { F } [ t ] \text { 多项式的空间 } t \text { 超过 } \mathbb { F } \text { . }

## 数学代写|线性代数代写linear algebra代考|R and C in Linear Algebra

$$\operatorname{dim} \mathbb{R} V=2 \operatorname{dim} \mathbb{C} V .$$

$$z_1 \mathbf{b}_1+\cdots+z_k \mathbf{b}_k=\mathbf{0},$$

$$z_1(1, i)+z_2(1,-1)=\left(z_1+z_2, i z_1-z_2\right)$$

$$z_1+z_2=0 i z_1-z_2=0 .$$

## 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。