#### Doug I. Jones

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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
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## 数学代写|高等线性代数代写Advanced Linear Algebra代考|Inner products and norms

Let $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$. We will write most results for the choice $\mathbb{F}=\mathbb{C}$. To interpret these results for the choice $\mathbb{F}=\mathbb{R}$, one simply ignores the complex conjugates that are part of the definitions.
Let $V$ be a vector space over $\mathbb{F}$. A function
$$\langle\cdot, \cdot\rangle: V \times V \rightarrow \mathbb{F}$$
is called a Hermitian form if
(i) $\langle\mathbf{x}+\mathbf{y}, \mathbf{z}\rangle=\langle\mathbf{x}, \mathbf{z}\rangle+\langle\mathbf{y}, \mathbf{z}\rangle$ for all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in V$.

(ii) $\langle a \mathbf{x}, \mathbf{y}\rangle=a\langle\mathbf{x}, \mathbf{y}\rangle$ for all $\mathbf{x}, \mathbf{y} \in V$ and all $a \in \mathbb{F}$.
(iii) $\langle\mathbf{x}, \mathbf{y}\rangle=\overline{\langle\mathbf{y}, \mathbf{x}\rangle}$, for all $\mathbf{x}, \mathbf{y} \in V$.
Notice that (iii) implies that $\langle\mathbf{x}, \mathbf{x}\rangle \in \mathbb{R}$ for all $\mathbf{x} \in V$. In addition, (ii) implies that $\langle\mathbf{0}, \mathbf{y}\rangle=0$ for all $\mathbf{y} \in V$. Also, (ii) and (iii) imply that $\langle\mathbf{x}, a \mathbf{y}\rangle=\bar{a}\langle\mathbf{x}, \mathbf{y}\rangle$ for all $\mathbf{x}, \mathbf{y} \in V$ and all $a \in \mathbb{F}$. Finally, (i) and (iii) imply that $\langle\mathbf{x}, \mathbf{y}+\mathbf{z}\rangle=\langle\mathbf{x}, \mathbf{y}\rangle+\langle\mathbf{x}, \mathbf{z}\rangle$ for all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in V$. Also,
The Hermitian form $\langle\cdot, \cdot\rangle$ is called an inner product if in addition
(iv) $\langle\mathbf{x}, \mathbf{x}\rangle>0$ for all $\mathbf{0} \neq \mathbf{x} \in V$.
If $V$ has an inner product $\langle\cdot, \cdot\rangle$ (or sometimes we say ” $V$ is endowed with the inner product $\langle\cdot, \cdot\rangle “)$, then we call the pair $(V,\langle\cdot, \cdot\rangle)$ an inner product space. At times we do not explicitly mention the inner product $\langle\cdot, \cdot\rangle$, and we refer to $V$ as an inner product space. In the latter case it is implicitly understood what the underlying inner product is, and typically it would be one of the standard inner products which we will encounter below.

## 数学代写|高等线性代数代写Advanced Linear Algebra代考|Orthogonal and orthonormal sets and bases

When a vector space has an inner product, it is natural to study objects that behave nicely with respect to the inner product. For bases this leads to the notions of orthogonality and orthonormality.

Given is an inner product space $(V,\langle\cdot, \cdot\rangle)$. When in an inner product space a norm $|\cdot|$ is used, then this norm is by default the associated norm $|\cdot|=\sqrt{\langle\cdot, \cdot\rangle}$ unless stated otherwise. We say that $\mathbf{v}$ and $\mathbf{w}$ are orthogonal if $\langle\mathbf{v}, \mathbf{w}\rangle=0$, and we will denote this as $\mathbf{v} \perp \mathbf{w}$. Notice that $\mathbf{0}$ is orthogonal to any vector, and it is the only vector that is orthogonal to itself.
For $\emptyset \neq W \subseteq V$ we define
$W^{\perp}={\mathbf{v} \in V:\langle\mathbf{v}, \mathbf{w}\rangle=0$ for all $\mathbf{w} \in W}={\mathbf{v} \in V: \mathbf{v} \perp \mathbf{w}$ for all $\mathbf{w} \in W}$
Notice that in this definition we do not require that $W$ is a subspace, thus $W$ can be any set of vectors of $V$.
Lemma 5.2.1 For $\emptyset \neq W \subseteq V$ we have that $W^{\perp}$ is a subspace of $V$.
Proof. Clearly $\mathbf{0} \in W^{\perp}$ as $\mathbf{0}$ is orthogonal to any vector, in particular to those in $W$. Next, let $\mathbf{x}, \mathbf{y} \in W^{\perp}$ and $c \in \mathbb{F}$. Then for every $\mathbf{w} \in W$ we have that $\langle\mathbf{x}+\mathbf{y}, \mathbf{w}\rangle=\langle\mathbf{x}, \mathbf{w}\rangle+\langle\mathbf{y}, \mathbf{w}\rangle=0+0=0$, and $\langle c \mathbf{x}, \mathbf{w}\rangle=c\langle\mathbf{x}, \mathbf{w}\rangle=c 0=0$. Thus $\mathbf{x}+\mathbf{y}, c \mathbf{x} \in W^{\perp}$, showing that $W^{\perp}$ is a subspace.
In Exercise 5.7.4 we will see that in case $W$ is a subspace of a finite-dimensional space $V$, then
$$\operatorname{dim} W+\operatorname{dim} W^{\prime}=\operatorname{dim} V .$$

# 高等线性代数代考

. .数学代写|

$$\langle\cdot, \cdot\rangle: V \times V \rightarrow \mathbb{F}$$(i) $\langle\mathbf{x}+\mathbf{y}, \mathbf{z}\rangle=\langle\mathbf{x}, \mathbf{z}\rangle+\langle\mathbf{y}, \mathbf{z}\rangle$对于所有$\mathbf{x}, \mathbf{y}, \mathbf{z} \in V$

，则称为厄米形式

(ii) $\langle a \mathbf{x}, \mathbf{y}\rangle=a\langle\mathbf{x}, \mathbf{y}\rangle$ 为所有人 $\mathbf{x}, \mathbf{y} \in V$ 所有的 $a \in \mathbb{F}$.
(iii) $\langle\mathbf{x}, \mathbf{y}\rangle=\overline{\langle\mathbf{y}, \mathbf{x}\rangle}$，对所有人来说 $\mathbf{x}, \mathbf{y} \in V$.

(iv)则称为内积 $\langle\mathbf{x}, \mathbf{x}\rangle>0$ 为所有人 $\mathbf{0} \neq \mathbf{x} \in V$.

## 数学代写|高等线性代数代写高级线性代数代考|正交和标准正交集和基

For $\emptyset \neq W \subseteq V$ 我们定义
$W^{\perp}={\mathbf{v} \in V:\langle\mathbf{v}, \mathbf{w}\rangle=0$ 为所有人 $\mathbf{w} \in W}={\mathbf{v} \in V: \mathbf{v} \perp \mathbf{w}$ 为所有人 $\mathbf{w} \in W}$注意，在这个定义中我们不要求这样做 $W$ 是子空间吗 $W$ 可以是的向量的任何集合 $V$

$$\operatorname{dim} W+\operatorname{dim} W^{\prime}=\operatorname{dim} V .$$

## 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。

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