### 数学代写|计算线性代数代写Computational Linear Algebra代考|Linear Systems

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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|计算线性代数代写Computational Linear Algebra代考|Basic Properties

A linear system has a unique solution, infinitely many solutions, or no solution. To discuss this we first consider the real case, and a homogeneous underdetermined system.

Lemma $1.3$ (Underdetermined System) Suppose $A \in \mathbb{R}^{m \times n}$ with $m<n$. Then there is a nonzero $\boldsymbol{x} \in \mathbb{R}^{n}$ such that $A \boldsymbol{x}=\mathbf{0}$.

Proof Suppose $A \in \mathbb{R}^{m \times n}$ with $m<n$. The $n$ columns of $A$ span a subspace of $\mathbb{R}^{m}$. Since $\mathbb{R}^{m}$ has dimension $m$ the dimension of this subspace is at most $m$. By Lemma $1.1$ the columns of $A$ must be linearly dependent. It follows that there is a nonzero $\boldsymbol{x} \in \mathbb{R}^{n}$ such that $\boldsymbol{A x}=\mathbf{0}$.
A square matrix is either nonsingular or singular.
Definition $1.7$ (Real Nonsingular or Singular Matrix) A square matrix $A \in$ $\mathbb{R}^{n \times n}$ is said to be nonsingular if the only real solution of the homogeneous system $A \boldsymbol{x}=\mathbf{0}$ is $\boldsymbol{x}=\mathbf{0}$. The matrix is singular if there is a nonzero $\boldsymbol{x} \in \mathbb{R}^{n}$ such that $A x=0$.

Theorem $1.6$ (Linear Systems; Existence and Uniqueness) Suppose $A \in \mathbb{R}^{n \times n}$. The linear system $\boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}$ has a unique solution $\boldsymbol{x} \in \mathbb{R}^{n}$ for any $\boldsymbol{b} \in \mathbb{R}^{n}$ if and only if the matrix $A$ is nonsingular.

Proof Suppose $\boldsymbol{A}$ is nonsingular. We define $\boldsymbol{B}=\left[\begin{array}{ll}\boldsymbol{a}\end{array}\right] \in \mathbb{R}^{n \times(n+1)}$ by adding a column to $\boldsymbol{A}$. By Lemma $1.3$ there is a nonzero $z \in \mathbb{R}^{n+1}$ such that $\boldsymbol{B} z=\mathbf{0}$. If we write $z=\left[\begin{array}{c}\bar{z} \ z_{n+1}\end{array}\right]$ where $\bar{z}=\left[z_{1}, \ldots, z_{n}\right]^{T} \in \mathbb{R}^{n}$ and $z_{n+1} \in \mathbb{R}$, then
$$\boldsymbol{B} z=\left[\begin{array}{ll} \boldsymbol{b} \end{array}\right]\left[\begin{array}{c} \bar{z} \ z_{n+1} \end{array}\right]=\boldsymbol{A} \bar{z}+z_{n+1} \boldsymbol{b}=\mathbf{0}$$

## 数学代写|计算线性代数代写Computational Linear Algebra代考|The Inverse Matrix

Suppose $\boldsymbol{A} \in \mathbb{C}^{n \times n}$ is a square matrix. A matrix $\boldsymbol{B} \in \mathbb{C}^{n \times n}$ is called a right inverse of $\boldsymbol{A}$ if $\boldsymbol{A} \boldsymbol{B}=\boldsymbol{I}$. A matrix $C \in \mathbb{C}^{n \times n}$ is said to be a left inverse of $A$ if $\boldsymbol{C}=\boldsymbol{I}$. We say that $A$ is invertible if it has both a left- and a right inverse. If $A$ has a right inverse $\boldsymbol{B}$ and a left inverse $\boldsymbol{C}$ then
$$C=C I=C(\boldsymbol{A} \boldsymbol{B})=(\boldsymbol{C} \boldsymbol{A}) \boldsymbol{B}=\boldsymbol{I} \boldsymbol{B}=\boldsymbol{B}$$
and this common inverse is called the inverse of $A$ and denoted by $A^{-1}$. Thus the inverse satisfies $A^{-1} A=A A^{-1}=I$.

We want to characterize the class of invertible matrices and start with a lemma.
Theorem $1.8$ (Product of Nonsingular Matrices) If $\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} \in \mathbb{C}^{n \times n}$ with $\boldsymbol{A} \boldsymbol{B}=$ $\boldsymbol{C}$ then $\boldsymbol{C}$ is nonsingular if and only if both $\boldsymbol{A}$ and $\boldsymbol{B}$ are nonsingular. In particular, if either $\boldsymbol{A} \boldsymbol{B}=\boldsymbol{I}$ or $\boldsymbol{B} \boldsymbol{A}=\boldsymbol{I}$ then $\boldsymbol{A}$ is nonsingular and $\boldsymbol{A}^{-1}=\boldsymbol{B}$.

Proof Suppose both $\boldsymbol{A}$ and $\boldsymbol{B}$ are nonsingular and let $\boldsymbol{C} \boldsymbol{x}=\mathbf{0}$. Then $\boldsymbol{A} \boldsymbol{B x}=\mathbf{0}$ and since $\boldsymbol{A}$ is nonsingular we see that $\boldsymbol{B} \boldsymbol{x}=\mathbf{0}$. Since $\boldsymbol{B}$ is nonsingular we have $\boldsymbol{x}=\mathbf{0}$. We conclude that $C$ is nonsingular.

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Determinants

The first systematic treatment of determinants was given by Cauchy in 1812 . He adopted the word “determinant”. The first use of determinants was made by Leibniz in 1693 in a letter to De L’Hôspital. By the beginning of the twentieth century the theory of determinants filled four volumes of almost 2000 pages (Muir, 1906-1923. Historic references can be found in this work). The main use of determinants in this text will be to study the characteristic polynomial of a matrix and to show that a matrix is nonsingular.
For any $A \in \mathbb{C}^{n \times n}$ the determinant of $A$ is defined by the number
$$\operatorname{det}(\boldsymbol{A})=\sum_{\sigma \in S_{n}} \operatorname{sign}(\sigma) a_{\sigma(1), 1} a_{\sigma(2), 2} \cdots a_{\sigma(n), n}$$
This sum ranges of all $n$ ! permutations of ${1,2, \ldots, n}$. Moreover, $\operatorname{sign}(\sigma)$ equals the number of times a bigger integer precedes a smaller one in $\sigma$. We also denote the determinant by (Cayley, 1841)
$$\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \vdots & \vdots & & \vdots \ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|$$
From the definition we have
$$\left|\begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right|=a_{11} a_{22}-a_{21} a_{12}$$
The first term on the right corresponds to the identity permutation $\epsilon$ given by $\epsilon(i)=$ $i, i=1,2$. The second term comes from the permutation $\sigma={2,1}$. For $n=3$ there are six permutations of ${1,2,3}$. Then
$\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33}\end{array}\right|=a_{11} a_{22} a_{33}-a_{11} a_{32} a_{23}-a_{21} a_{12} a_{33}$
$+a_{21} a_{32} a_{13}+a_{31} a_{12} a_{23}-a_{31} a_{22} a_{13}$
This follows since $\operatorname{sign}({1,2,3})=\operatorname{sign}({2,3,1})=\operatorname{sign}({3,1,2})=1$, and noting that interchanging two numbers in a permutation reverses it sign we find $\operatorname{sign}({2,1,3})=\operatorname{sign}({3,2,1})=\operatorname{sign}({1,3,2})=-1$.

## 数学代写|计算线性代数代写Computational Linear Algebra代考|The Inverse Matrix

C=C我=C(一个乙)=(C一个)乙=我乙=乙

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Determinants

1812 年，Cauchy 首次系统地处理了行列式。他采用了“决定因素”一词。1693 年莱布尼茨在给 De L’Hôspital 的一封信中首次使用了行列式。到 20 世纪初，行列式理论占据了将近 2000 页的四卷本（缪尔，1906-1923 年。可以在这部著作中找到历史参考资料）。本文中行列式的主要用途是研究矩阵的特征多项式并证明矩阵是非奇异的。

|一个11一个12⋯一个1n 一个21一个22⋯一个2n ⋮⋮⋮ 一个n1一个n2⋯一个nn|

|一个11一个12 一个21一个22|=一个11一个22−一个21一个12

|一个11一个12一个13 一个21一个22一个23 一个31一个32一个33|=一个11一个22一个33−一个11一个32一个23−一个21一个12一个33
+一个21一个32一个13+一个31一个12一个23−一个31一个22一个13

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。