• 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|高等线性代数代写Advanced Linear Algebra代考|Linear independence, span, and basis

The notion of a basis is a crucial one; it basically singles out few elements in the vector space with which we can reconstruct the whole vector space. For example, the monomials $1, X, X^2, \ldots$ form a basis of the vector space of polynomials. When we start to do certain (namely, linear) operations on elements of a vector space, we will see in the next chapter that it will suffice to know how these operations act on the basis elements. Differentiation is an example: as soon as we know that the derivatives of $1, X, X^2, X^3, \ldots$ are $0,1,2 X, 3 X^2, \ldots$, respectively, it is easy to find the derivative of a polynomial. Before we get to the notion of a basis, we first need to introduce linear independence and span.

Let $V$ be a vector space over $\mathbb{F}$. A set of vectors $\left{\mathbf{v}_1, \ldots, \mathbf{v}_p\right}$ in $V$ is said to be linearly independent if the vector equation
$$c_1 \mathbf{v}_1+c_2 \mathbf{v}_2+\cdots+c_p \mathbf{v}_p=\mathbf{0},$$
with $c_1, \ldots, c_p \in \mathbb{F}$, only has the solution $c_1=0, \ldots, c_p=0$ (the trivial solution). The set $\left{\mathbf{v}_1, \ldots, \mathbf{v}_p\right}$ is said to be linearly dependent if (2.2) has a solution where not all of $c_1, \ldots, c_p$ are zero (a nontrivial solution). In such a case, (2.2) with at least one $c_i$ nonzero gives a linear dependence relation among $\left{\mathbf{v}_1, \ldots, \mathbf{v}_p\right}$. An arbitrary set $S \subseteq V$ is said to be linearly independent if everry finitè subsét of $S$ is linearly indēpendent. Thẻ set $S$ is linearly dependent, if it is not linearly independent.

We will see in this section that any n-dimensional vector space over $\mathbb{F}$ “works the same” as $\mathbb{F}^n$, which simplifies the study of such vector spaces tremendously. To make this idea more precise, we have to discuss coordinate systems. We start with the following result.

Theorem 2.5.1 Let $\mathcal{B}=\left{\mathbf{v}_1, \ldots, \mathbf{v}_n\right}$ be a basis for a vector space $V$ over $\mathbb{F}$. Then for each $\mathbf{v} \in V$ there exists unique $c_1, \ldots, c_n \in \mathbb{F}$ so that
$$\mathbf{v}=c_1 \mathbf{v}_1+\cdots+c_n \mathbf{v}_n .$$
Proof. Let $\mathbf{v} \in V$. As $\operatorname{Span} \mathcal{B}=V$, we have that $\mathbf{v}=c_1 \mathbf{v}_1+\cdots+c_n \mathbf{v}_n$ for some $c_1, \ldots, c_n \in \mathbb{F}$. Suppose that we also have $\mathbf{v}=d_1 \mathbf{v}_1+\cdots+d_n \mathbf{v}_n$ for some $d_1, \ldots, d_n \in \mathbb{F}$. Then
$$\mathbf{0}=\mathbf{v}-\mathbf{v}=\sum_{j=1}^n c_j \mathbf{v}j-\sum{j=1}^n d_j \mathbf{v}_j=\left(c_1-d_1\right) \mathbf{v}_1+\cdots+\left(c_n-d_n\right) \mathbf{v}_n .$$
As $\left{\mathbf{v}_1, \ldots, \mathbf{v}_n\right}$ is linearly independent, we must have $c_1-d_1=0, \ldots, c_n-d_n=0$. This yields $c_1=d_1, \ldots, c_n=d_n$, yielding the uniqueness.

When ( $2.9)$ holds, we say that $c_1, \ldots, c_n$ are the coordinates of $\mathbf{v}$ relative to the basis $\mathcal{B}$, and we write
$$[\mathbf{v}]{\mathcal{B}}=\left(\begin{array}{c} c_1 \ \vdots \ c_n \end{array}\right) .$$ Thus, when $\mathcal{B}=\left{\mathbf{v}_1, \ldots, \mathbf{v}_n\right}$ we have $$\mathbf{v}=c_1 \mathbf{v}_1+\cdots+c_n \mathbf{v}_n \Leftrightarrow[\mathbf{v}]{\mathcal{B}}=\left(\begin{array}{c} c_1 \ \vdots \ c_n \end{array}\right)$$

# 高等线性代数代考

## 数学代写|高等线性代数代写Advanced Linear Algebra代考|Linear independent, span, and basis

.

$$c_1 \mathbf{v}_1+c_2 \mathbf{v}_2+\cdots+c_p \mathbf{v}_p=\mathbf{0},$$

. 在本节中，我们将看到$\mathbb{F}$上的任何n维向量空间与$\mathbb{F}^n$“工作原理相同”，这极大地简化了对此类向量空间的研究。为了使这个想法更精确，我们必须讨论坐标系。我们从以下结果开始:

$$\mathbf{v}=c_1 \mathbf{v}_1+\cdots+c_n \mathbf{v}_n .$$

$$\mathbf{0}=\mathbf{v}-\mathbf{v}=\sum_{j=1}^n c_j \mathbf{v}j-\sum{j=1}^n d_j \mathbf{v}_j=\left(c_1-d_1\right) \mathbf{v}_1+\cdots+\left(c_n-d_n\right) \mathbf{v}_n .$$

$$[\mathbf{v}]{\mathcal{B}}=\left(\begin{array}{c} c_1 \ \vdots \ c_n \end{array}\right) .$$因此，当$\mathcal{B}=\left{\mathbf{v}_1, \ldots, \mathbf{v}_n\right}$我们有$$\mathbf{v}=c_1 \mathbf{v}_1+\cdots+c_n \mathbf{v}_n \Leftrightarrow[\mathbf{v}]{\mathcal{B}}=\left(\begin{array}{c} c_1 \ \vdots \ c_n \end{array}\right)$$

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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。

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