text: Posted on Categories:Mathematical Analysis, 数学代写, 数学分析
  
  # 数学代写|数学分析作业代写Mathematical Analysis代考|Definitions and Basic Properties
  
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  ## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties
  
  This section is a summary of the most basic concepts of vector space theory. The main reason for including this section is to establish terminology and provide a collection of important examples. The reader should pay particular attention to the examples, because the sequence and function spaces we introduce here are of fundamental importance for the rest of the book. The theorems are stated without proof.
  
  Definition. Let $\mathbb{K}$ be a field, and suppose $U$ is a nonempty set equipped with a binary operation, + (vector addition). Suppose also that there is a function $\times: \mathbb{K} \times U \rightarrow U$ (scalar multiplication) that assigns to each pair $(a, u) \in \mathbb{K} \times U$ an element $a \times u$ (or simply $a u$ ) in $U$. The triple $(U,+, x)$ is called a vector space over the field $\mathbb{K}$ if the following conditions are satisfied by all elements $a, b \in \mathbb{K}$ and all elements $u, v, w \in U:$
  (a) $u+v=v+u$
  (b) $u+(v+w)=(u+v)+w$.
  (c) There is an element $0 \in V$ (the zero vector) such that $u+0=u$.
  (d) For every $u \in U$, there is an element $-u \in U$ such that $u+(-u)=0$.
  (e) $a(u+v)=a u+a v$.
  (f) $(a+b) u=a u+b u$.
  (g) $(a b) u=a(b u)$.
  (h) $1 . u=u$.
  
  ## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Independent Sets and Bases
  
  This section is focused on the concepts on linear independence and bases. Our approach to studying bases is unified in the sense that we do not treat finitedimensional and infinite-dimensional spaces separately. We use Zorn’s lemma to prove the existence of a basis. A number of important equivalent characterizations of a basis are also discussed, both in the body of the section as well as in the section exercises.
  
  Definition. A finite subset $\left{u_1, u_2, \ldots, u_n\right}$ of a vector space $U$ is dependent if there exist scalars $a_1, a_2, \ldots, a_n$, not all zero, such that $\sum_{i=1}^n a_i u_i=0$.
  
  Terminology. A vector of the form $\sum_{i=1}^n a_i u_i$, where at least one $a_i \neq 0$, is called a nontrivial linear combination of $u_1, u_2, \ldots, u_n$. The above definition can be restated as follows: $\left{u_1, u_2, \ldots, u_n\right}$ is dependent if some nontrivial linear combination of $u_1, u_2, \ldots, u_n$ is zero.
  
  Theorem 3.2.1. A subset $S=\left{u_1, u_2, \ldots, u_n\right}$ of a vector space $U$ is dependent if and only if one of the vectors in $S$ is a linear combination of the remaining vectors.
  Proof. Suppose $\left{u_1, u_2, \ldots, u_n\right}$ is dependent. Then $\sum_{i=1}^n a_i u_i=0$ for scalars $a_1, a_2, \ldots, a_n$, not all zero. Say $a_i \neq 0$. Then $u_i=\frac{-1}{a_i} \sum_{j \neq i}^n a_j u_j$. Conversely, if $u_i=\sum_{j \neq i}^n a_j u_j$, then $1 . u_i-\sum_{j \neq i}^n a_j u_j=0$, and $\left{u_1, u_2, \ldots, u_n\right}$ is dependent.
  
  ## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties
  
  (a) $u+v=v+u$
  (b) $u+(v+w)=(u+v)+w$。
  (c)有一个元素$0 \in V$(零向量)使得$u+0=u$。
  (d)对于每一个$u \in U$，都有一个元素$-u \in U$，使得$u+(-u)=0$。
  (e) $a(u+v)=a u+a v$。
  (f) $(a+b) u=a u+b u$。
  (g) $(a b) u=a(b u)$。
  (h) $1 . u=u$。
  
  ## 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。
explanation: The provided text is an excerpt from a textbook or course material on Mathematical Analysis, specifically covering the topics of vector spaces and linear independence. The text introduces definitions and basic properties of vector spaces, including the axioms that must be satisfied for a set to be considered a vector space over a field. It also delves into the concepts of linear independence and bases, providing a theorem and proof related to dependent sets.
  
  The text does not contain a specific question or answer but rather presents definitions, theorems, and proofs as part of a broader educational exposition.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Posted on Categories:Mathematical Analysis, 数学代写, 数学分析
  
  # 数学代写|数学分析作业代写Mathematical Analysis代考|Definitions and Basic Properties
  
  avatest™
  
  ## avatest™帮您通过考试
  
  avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！
  
  •最快12小时交付
  
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  •70分以下全额退款
  
  ## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties
  
  This section is a summary of the most basic concepts of vector space theory. The main reason for including this section is to establish terminology and provide a collection of important examples. The reader should pay particular attention to the examples, because the sequence and function spaces we introduce here are of fundamental importance for the rest of the book. The theorems are stated without proof.
  
  Definition. Let $\mathbb{K}$ be a field, and suppose $U$ is a nonempty set equipped with a binary operation, + (vector addition). Suppose also that there is a function $\times: \mathbb{K} \times U \rightarrow U$ (scalar multiplication) that assigns to each pair $(a, u) \in \mathbb{K} \times U$ an element $a \times u$ (or simply $a u$ ) in $U$. The triple $(U,+, x)$ is called a vector space over the field $\mathbb{K}$ if the following conditions are satisfied by all elements $a, b \in \mathbb{K}$ and all elements $u, v, w \in U:$
  (a) $u+v=v+u$
  (b) $u+(v+w)=(u+v)+w$.
  (c) There is an element $0 \in V$ (the zero vector) such that $u+0=u$.
  (d) For every $u \in U$, there is an element $-u \in U$ such that $u+(-u)=0$.
  (e) $a(u+v)=a u+a v$.
  (f) $(a+b) u=a u+b u$.
  (g) $(a b) u=a(b u)$.
  (h) $1 . u=u$.
  
  ## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Independent Sets and Bases
  
  This section is focused on the concepts on linear independence and bases. Our approach to studying bases is unified in the sense that we do not treat finitedimensional and infinite-dimensional spaces separately. We use Zorn’s lemma to prove the existence of a basis. A number of important equivalent characterizations of a basis are also discussed, both in the body of the section as well as in the section exercises.
  
  Definition. A finite subset $\left{u_1, u_2, \ldots, u_n\right}$ of a vector space $U$ is dependent if there exist scalars $a_1, a_2, \ldots, a_n$, not all zero, such that $\sum_{i=1}^n a_i u_i=0$.
  
  Terminology. A vector of the form $\sum_{i=1}^n a_i u_i$, where at least one $a_i \neq 0$, is called a nontrivial linear combination of $u_1, u_2, \ldots, u_n$. The above definition can be restated as follows: $\left{u_1, u_2, \ldots, u_n\right}$ is dependent if some nontrivial linear combination of $u_1, u_2, \ldots, u_n$ is zero.
  
  Theorem 3.2.1. A subset $S=\left{u_1, u_2, \ldots, u_n\right}$ of a vector space $U$ is dependent if and only if one of the vectors in $S$ is a linear combination of the remaining vectors.
  Proof. Suppose $\left{u_1, u_2, \ldots, u_n\right}$ is dependent. Then $\sum_{i=1}^n a_i u_i=0$ for scalars $a_1, a_2, \ldots, a_n$, not all zero. Say $a_i \neq 0$. Then $u_i=\frac{-1}{a_i} \sum_{j \neq i}^n a_j u_j$. Conversely, if $u_i=\sum_{j \neq i}^n a_j u_j$, then $1 . u_i-\sum_{j \neq i}^n a_j u_j=0$, and $\left{u_1, u_2, \ldots, u_n\right}$ is dependent.
  
  ## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Definitions and Basic Properties
  
  (a) $u+v=v+u$
  (b) $u+(v+w)=(u+v)+w$。
  (c)有一个元素$0 \in V$(零向量)使得$u+0=u$。
  (d)对于每一个$u \in U$，都有一个元素$-u \in U$，使得$u+(-u)=0$。
  (e) $a(u+v)=a u+a v$。
  (f) $(a+b) u=a u+b u$。
  (g) $(a b) u=a(b u)$。
  (h) $1 . u=u$。
  
  ## 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。
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an excerpt from a textbook or course material on Mathematical Analysis, specifically covering the topics of vector spaces and linear independence. The text introduces definitions and basic properties of vector spaces, including the axioms that must be satisfied for a set to be considered a vector space over a field. It also delves into the concepts of linear independence and bases, providing a theorem and proof related to dependent sets.
  
  The text does not contain a specific question or answer but rather presents definitions, theorems, and proofs as part of a broader educational exposition.
  
  Label: no-question