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statistics-lab™ 为您的留学生涯保驾护航 在代写计算线性代数Computational 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。我们的专家在代写计算线性代数Computational Linear Algebra代写方面经验极为丰富，各种代写计算线性代数Computational 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|计算线性代数代写Computational Linear Algebra代考|Partial Fractions

Definition 5.20. Let $p_{1}(z)=\sum_{k=0}^{m} a_{k} z^{k}$ and $p_{2}(z)=\sum_{k=0}^{n} b_{k} z^{k}$ be two complex polynomials. The function $Q(z)$ obtained by dividing $p_{1}(z)$ by $p_{2}(z)$,
$$Q(z)=\frac{p_{1}(z)}{p_{2}(z)}$$
is said rational fraction in the variable $z$.
Let $Q(z)=\frac{p_{1}(z)}{p_{2}(z)}$ be a rational fraction in the variable $z$. The partial fraction decomposition (or partial fraction expansion) is a mathematical procedure that consists of expressing the fraction as a sum of rational fractions where the denominators are of lower order that that of $p_{2}(z)$ :
$$Q(z)=\frac{p_{1}(z)}{p_{2}(z)}=\sum_{k=1}^{n} \frac{f_{i}(z)}{g_{i}(z)}$$
This decomposition can be of great help to break a complex problem into many simple problems. For example, the integration term by term can be much easier if the fraction has been decomposed, see Chap. $13 .$

Let us consider the case of a proper fraction, i.e. the order of $p_{1}(z) \leq$ than the order of $p_{2}(z)(m \leq n)$. Let us indicate with the term zeros the values $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}$ such that $p_{1}\left(\alpha_{k}\right)=0 \forall k \in \mathbb{N}$ with $1 \leq k \leq n$, and the term poles the values $\beta_{1}, \beta_{2}, \ldots, \beta_{n}$ such that $p_{2}\left(\beta_{k}\right)=0 \forall k \in \mathbb{N}$ with $1 \leq k \leq n$.
Let us distinguish three cases:

• rational fractions with distinct/single real or complex poles
• rational fractions with multiple real or complex poles
• rational fractions with conjugate complex poles
Rational fractions with only distinct poles are characterized by a denominator of the kind
$$p_{2}(z)-\left(z-\beta_{1}\right)\left(z-\beta_{2}\right) \ldots\left(z-\beta_{n}\right)$$

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Basic Concepts: Lines in the Plane

This chapter introduces the conics and characterizes them from an algebraic perspective. While in depth geometrical aspects of the conics lie outside the scopes of this chapter, this chapter is an opportunity to revisit concepts studied in other chapters such as matrix and determinant and assign a new geometric characterization to them.

In order to achieve this aim, let us start with considering the three-dimensional space. Intuitively, we may think that, within this space, points, lines, and planes exist.

We have previously introduced, in Chap. 4, the concept of line as representation of the set $\mathbb{R}$. If $\mathbb{R}^{2}$ can be represented as the plane, a line is an infinite subset of $\mathbb{R}^{2}$. We have also introduced in Chap. 4 the concepts of point, distance between two points, segment, and direction of a line. From the algebra of the vectors we also know that the direction of a line is identified by the components of a vector having the same direction, i.e. a line can be characterized by two numbers which we will indicate here as $(l, m)$.

Definition 6.1. Let $\mathbf{P}$ and $\mathbf{Q}$ be two points of the plane and $d \overline{\mathrm{PQ}}$ be the distance between two points. The point $\mathbf{M}$ of the segment $\overline{\mathbf{P Q}}$ such that $d_{\overline{\mathbf{P M}}}=d_{\overline{\mathbf{M Q}}}$ is said middle point.

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

Let $l_{1}$ and $l_{2}$ be two lines of the plane having equation, respectively,
\begin{aligned} &l_{1}: a_{1} x+b_{1} y+c_{1}=0 \ &l_{2}: a_{2} x+b_{2} y+c_{2}=0 \end{aligned}
We aim at studying the position of these two line in the plane. If these two line intersect in a point $\mathbf{P}{\mathbf{0}}$ it follows that the point $\mathbf{P}{\mathbf{0}}$ belongs to both the line. Equivalently we may state that the coordinates $\left(x_{0}, y_{0}\right)$ of this point $\mathbf{P}{\mathbf{0}}$ simultaneously satisfy the equations of the lines $l{1}$ and $l_{2}$.

In other words, $\left(x_{0}, y_{0}\right)$ is the solution of the following system of linear equations:
$$\left{\begin{array}{l} a_{1} x+b_{1} y+c_{1}=0 \ a_{2} x+b_{2} y+c_{2}=0 \end{array}\right.$$

At first, we may observe that a new characterization of the concept of system of linear equations is given. A system of linear equation can be seen as a set of lines and its solution, when it exists, is the intersection of these lines. In this chapter we study lines in the plane. Thus, the system has two linear equations in two variables. A system having size $3 \times 3$ can be seen as the equation of three lines in the space. By extension an $n \times n$ system of linear equation represents lines in a $n$-dimensional space. In general, even when not all the equations are equations of the line, the solutions of a system of equations can be interpreted as the intersection of objects.
Let us focus on the case of two lines in the plane. The system above is associated with the following incomplete and complete matrices, respectively,
$$\mathbf{A}=\left(\begin{array}{ll} a_{1} & b_{1} \ a_{2} & b_{2} \end{array}\right)$$
and
$$\mathbf{A}^{\mathbf{c}}=\left(\begin{array}{lll} a_{1} & b_{1} & -c_{1} \ a_{2} & b_{2} & -c_{2} \end{array}\right)$$

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

$$Q(z)=\frac{p_{1}(z)}{p_{2}(z)}$$

$$Q(z)=\frac{p_{1}(z)}{p_{2}(z)}=\sum_{k=1}^{n} \frac{f_{i}(z)}{g_{i}(z)}$$

• 具有不同/单个实数或复数极点的有理分数
• 具有多个实极或复极点的有理分数
• 具有共轭复极点
的有理分数只有不同极点的有理分数的特征是分母
$$p_{2}(z)-\left(z-\beta_{1}\right)\left(z-\beta_{2}\right) \ldots\left(z-\beta_{n}\right)$$

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

$$l_{1}: a_{1} x+b_{1} y+c_{1}=0 \quad l_{2}: a_{2} x+b_{2} y+c_{2}=0$$

$\$ \$$Veft {$$
a_{1} x+b_{1} y+c_{1}=0 a_{2} x+b_{2} y+c_{2}=0
$$正确的。 \ \$$

$$\mathbf{A}^{\mathbf{c}}=\left(\begin{array}{lllll} a_{1} & b_{1} & -c_{1} a_{2} & b_{2} & -c_{2} \end{array}\right)$$

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师