### 数学代写|计算线性代数代写Computational Linear Algebra代考|MAST10007

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|A Preliminary Introduction to Algebraic Structures

If a set is a primitive concept, on the basis of a set, algebraic structures are sets that allow some operations on their elements and satisfy some properties. Although an in depth analysis of algebraic structures is out of the scopes of this chapter, this section gives basic definitions and concepts. More advanced concepts related to algebraic structures will be given in Chap. $7 .$

Definition 1.32. An operation is a function $f: A \rightarrow B$ where $A \subset X_{1} \times X_{2} \times \ldots \times X_{k}$, $k \in \mathbb{N}$. The $k$ value is said arity of the operation.

Definition 1.33. Let us consider a set $A$ and an operation $f: A \rightarrow B$. If $A$ is $X \times X \times$ $\ldots \times X$ and $B$ is $X$, i.e. the result of the operation is still a member of the set, the set is said to be closed with respect to the operation $f$.

Definition 1.34. Ring. A ring $R$ is a set equipped with two operations called sum and product. The sum is indicated with $\mathrm{a}+$ sign while the product operator is simply omitted (the product of $x_{1}$ by $x_{2}$ is indicated as $x_{1} x_{2}$ ). Both these operations process two elements of $R$ and return an element of $R(R$ is closed with respect to these two operations). In addition, the following properties must be valid.

• commutativity (sum): $x_{1}+x_{2}=x_{2}+x_{1}$
• associativity (sum): $\left(x_{1}+x_{2}\right)+x_{3}=x_{1}+\left(x_{2}+x_{3}\right)$
• neutral element (sum): $\exists$ an element $0 \in R$ such that $\forall x \in R: x+0=x$
• inverse element (sum): $\forall x \in R: \exists(-x) \mid x+(-x)=0$
• associativity (product): $\left(x_{1} x_{2}\right) x_{3}=x_{1}\left(x_{2} x_{3}\right)$
• distributivity $1: x_{1}\left(x_{2}+x_{3}\right)=x_{1} x_{2}+x_{1} x_{3}$
• distributivity $2:\left(x_{2}+x_{3}\right) x_{1}=x_{2} x_{1}+x_{3} x_{1}$
• neutral element (product): $\exists$ an element $1 \in R$ such that $\forall x \in R x 1=1 x=x$
The inverse element with respect to the sum is also named opposite element.

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

Although this chapter intentionally refers to the set of real numbers $\mathbb{R}$ and its sum and multiplication operations, all the concepts contained in this chapter can be easily extended to the set of complex numbers $\mathbb{C}$ and the complex field. This fact is further remarked in Chap. 5 after complex numbers and their operations are introduced.
Definition 2.1. Numeric Vector. Let $n \in \mathbb{N}$ and $n>0$. The set generated by the Cartesian product of $\mathbb{R}$ by itself $n$ times $(\mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \ldots)$ is indicated with $\mathbb{R}^{n}$ and is a set of ordered $n$-tuples of real numbers. The generic element $\mathbf{a}=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of this set is named numeric vector or simply vector of order $n$ on the real field and the generic $a_{i} \forall i$ from 1 to $n$ is said the $i^{t h}$ component of the vector a.
Example 2.1. The $n$-tuple
$$\mathbf{a}=(1,0,56.3, \sqrt{2})$$
is a vector of $\mathbb{R}^{4}$.
Definition 2.2. Scalar. A numeric vector $\lambda \in \mathbb{R}^{1}$ is said scalar.
Definition 2.3. Let $\mathbf{a}=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ and $\mathbf{b}=\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ be two numeric vectors $\in \mathbb{R}^{n}$. The sum of these two vectors is the vector $\mathbf{c}=\left(a_{1}+b_{1}, a_{2}+b_{2}, \ldots, a_{n}\right.$ $\left.+b_{n}\right)$ generated by the sum of the corresponding components.
Example 2.2. Let us consider the following vectors of $\mathbb{R}^{3}$
\begin{aligned} &\mathbf{a}=(1,0,3) \ &\mathbf{b}=(2,1,-2) \end{aligned}

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Basic Definitions About Matrices

Definition 2.6. Matrix. Let $m, n \in \mathbb{N}$ and both $m, n>0$. A matrix $(m \times n) \mathbf{A}$ is a generic table of the kind:
$$\boldsymbol{\Lambda}=\left(\begin{array}{cccc} a_{1,1} & a_{1,2} & \ldots & a_{1, n} \ a_{2,1} & a_{2,2} & \ldots & a_{2, n} \ \ldots & \ldots & \ldots & \ldots \ a_{m, 1} & a_{m, 2} & \ldots & a_{m, n} \end{array}\right)$$
where each matrix element $a_{i, j} \in \mathbb{R}$. If $m=n$ the matrix is said square while it is said rectangular otherwise.

The numeric vector $\mathbf{a}{\mathbf{i}}=\left(a{i, 1}, a_{i, 2}, \ldots, a_{i, n}\right)$ is said generic $i^{t h}$ row vector while $\mathbf{a}^{\mathbf{j}}=\left(a_{1, j}, a_{2, j}, \ldots, a_{m, j}\right)$ is said generic $j^{\text {th }}$ column vector.

The set containing all the matrices of real numbers having $m$ rows and $n$ columns is indicated with $\mathbb{R}{m, n}$. Definition 2.7. A matrix is said null $\mathbf{O}$ if all its elements are zeros. Example 2.5. The null matrix of $\mathbb{R}{2,3}$ is
$$\mathbf{O}=\left(\begin{array}{lll} 0 & 0 & 0 \ 0 & 0 & 0 \end{array}\right)$$
Definition 2.8. Let $\mathbf{A} \in \mathbb{R}{m, n}$. The transpose matrix of $\mathbf{A}$ is a matrix $\mathbf{A}^{\mathbf{T}}$ whose elements are the same of $\mathbf{A}$ but $\forall i, j: a{j, i}=a_{i, j}^{T}$.
Example 2.6.
$\mathbf{A}=\left(\begin{array}{cccc}2 & 7 & 3.4 & \sqrt{2} \ 5 & 0 & 4 & 1\end{array}\right)$
$\mathbf{A}^{\mathbf{T}}=\left(\begin{array}{cc}2 & 5 \ 7 & 0 \ 3.4 & 4 \ \sqrt{2} & 1\end{array}\right)$
It can be easily proved that the transpose of the transpose of a matrix is the matrix itself: $\left(\mathbf{A}^{\mathbf{T}}\right)^{\mathbf{T}}$.
Definition 2.9. A matrix $\mathbf{A} \in \mathbb{R}_{n, n}$ is said n order square matrix.

## 数学代写|计算线性代数代写Computational Linear Algebra代考|A Preliminary Introduction to Algebraic Structures

• 交换性 (总和) : $x_{1}+x_{2}=x_{2}+x_{1}$
• 关联性 (总和) : $\left(x_{1}+x_{2}\right)+x_{3}=x_{1}+\left(x_{2}+x_{3}\right)$
• 中性元素 (总和) : $\exists$ 个个元素 $0 \in R$ 这样 $\forall x \in R: x+0=x$
• 逆元素 (总和) : $\forall x \in R: \exists(-x) \mid x+(-x)=0$
• 关联性 (产品) : $\left(x_{1} x_{2}\right) x_{3}=x_{1}\left(x_{2} x_{3}\right)$
• 分配性 $1: x_{1}\left(x_{2}+x_{3}\right)=x_{1} x_{2}+x_{1} x_{3}$
• 分配性 $2:\left(x_{2}+x_{3}\right) x_{1}=x_{2} x_{1}+x_{3} x_{1}$
• 中性元素 (产品) : $\exists 一$ 个元素 $1 \in R$ 这样 $\forall x \in R x 1=1 x=x$ 关于和的逆元也称为逆元。

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

$$\mathbf{a}=(1,0,56.3, \sqrt{2})$$

$$\mathbf{a}=(1,0,3) \quad \mathbf{b}=(2,1,-2)$$

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Basic Definitions About Matrices

$$\mathbf{O}=\left(\begin{array}{llllll} 0 & 0 & 0 & 0 & 0 \end{array}\right)$$

$\mathbf{A}=\left(\begin{array}{lllllll}2 & 7 & 3.4 & \sqrt{2} 5 & 0 & 4 & 1\end{array}\right)$
$\mathbf{A}^{\mathbf{T}}=\left(\begin{array}{lllll}2 & 5 & 7 & 0 & 3.4\end{array}\right.$

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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。