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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

CS代写|计算机图形学作业代写computer graphics代考|Imaginary Numbers

Imaginary numbers were invented to resolve problems where an equation such as $x^2+16=0$, has no real solution (roots). The simple idea of declaring the existence of a quantity $i$, such that $i^2–1$, permits the solution to be expressed as
$$x=\pm 4 i .$$
For example, if $x=4 i$ we have
\begin{aligned} x^2+16 &=16 i^2+16 \ &=-16+16 \ &=0 \end{aligned}
and if $x=-4 i$ we have
\begin{aligned} x^2+16 &=16 i^2+16 \ &=-16+16 \ &=0 . \end{aligned}
But what is $i$ ? In 1637, the French mathematician René Descartes (1596-1650), published La Géométrie, in which he stated that numbers incorporating $\sqrt{-1}$ were ‘imaginary’, and for centuries this label has stuck. Unfortunately, it was a derogatory remark, as there is nothing ‘imaginary’ about $i$-it simply is an object that when introduced into various algebraic expressions, reveals some amazing underlying patterns. $i$ is not a number in the accepted sense, it is a mathematical object or construct that squares to $-1$. In some respects it is like time, which probably does not really exist, but is useful in describing the universe. However, $i$ does lose its mystery when interpreted as a rotational operator, which we investigate below.
As $i^2=-1$ then it must be possible to raise $i$ to other powers. For example,
$$i^4=i^2 i^2=1$$
and
$$i^5=i i^4=i .$$
Table $2.6$ shows the sequence up to $i^6$.

CS代写|计算机图形学作业代写computer graphics代考|Complex Numbers

A complex number has a real and imaginary part: $z=a+i b$, and represented by the set $\mathbb{C}$ :
$$z=a+b i \quad z \in \mathbb{C}, \quad a, b \in \mathbb{R}, \quad i^2=-1 .$$
Some examples are
\begin{aligned} &z=1+i \ &z=3-2 i \ &z=-23+\sqrt{23} i \end{aligned}
Complex numbers obey all the normal laws of algebra. For example, if we multiply $(a+b i)$ by $(c+d i)$ we have
$$(a+b i)(c+d i)=a c+a d i+b c i+b d i^2 .$$
Collecting up like terms and substituting $-1$ for $i^2$ we get
$$(a+b i)(c+d i)=a c+(a d+b c) i-b d$$
which simplifies to $$(a+b i)(c+d i)=a c-b d+(a d+b c) i$$
which is another complex number.
Something interesting happens when we multiply a complex number by its complex conjugate, which is the same complex number but with the sign of the imaginary part reversed:
$$(a+b i)(a-b i)=a^2-a b i+b a i-b^2 i^2 .$$
Collecting up like terms and simplifying we obtain
$$(a+b i)(a-b i)=a^2+b^2$$
which is a real number, as the imaginary part has been cancelled out by the action of the complex conjugate.

计算机图形学代考

CS代写|计算机图形学作业代写computer graphics代考|Imaginary Numbers

$$x^2+16=16 i^2+16=-16+16=0$$

$$x^2+16=16 i^2+16 \quad=-16+16=0 .$$

$$i^4=i^2 i^2=1$$

$$i^5=i i^4=i .$$

CS代写|计算机图形学作业代写computer graphics代考|Complex Numbers

$$z=a+b i \quad z \in \mathbb{C}, \quad a, b \in \mathbb{R}, \quad i^2=-1 .$$

$$z=1+i \quad z=3-2 i z=-23+\sqrt{23} i$$

$$(a+b i)(c+d i)=a c+a d i+b c i+b d i^2 .$$

$$(a+b i)(c+d i)=a c+(a d+b c) i-b d$$

$$(a+b i)(c+d i)=a c-b d+(a d+b c) i$$

$$(a+b i)(a-b i)=a^2-a b i+b a i-b^2 i^2 .$$

$$(a+b i)(a-b i)=a^2+b^2$$

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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。

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