## 数学代写|线性代数代写linear algebra代考|MATH1071

2022年12月26日

couryes-lab™ 为您的留学生涯保驾护航 在代写线性代数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。我们的专家在代写线性代数linear algebra代写方面经验极为丰富，各种代写线性代数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
couryes™为您提供可以保分的包课服务

## 数学代写|线性代数代写linear algebra代考|Subspaces of Rn

In Section 2.3, we introduced the $n$-dimensional Euclidean spaces $\mathbb{R}^n$. One nice aspect of these spaces is that they lend themselves to visualization: the space $\mathbb{R}=\mathbb{R}^1$ just looks like a number line, $\mathbb{R}^2$ is visualized as the $x y$-plane, and $\mathbb{R}^3$ can be represented with $x$-, $y$-, and $z$-axes as pictured in Figure 2.21. For larger $n$, the space is harder to visualize, because we would have to add more axes and the world we live in is very much like $\mathbb{R}^3$. As we continue our study of linear algebra, you will start to gain more intuition about how to think about these higher dimensional spaces.

In the last section we looked at a few examples of subspaces of $\mathbb{R}^n$. In this section we explore the geometry of such subspaces.
Theorem 2.5.16
Let $L$ be a line through the origin in $\mathbb{R}^n$. Then $L$ is a subspace of $\left(\mathbb{R}^n,+, \cdot\right)$.
Proof. A line through the origin in $\mathbb{R}^n$ is represented as the set $L={\beta v \mid \beta \in \mathbb{R}}$. We show $L$ is a subspace using the property given in Corollary 2.5.6. Let $w_1=\beta_1 v$ and $w_2=\beta_2 v$ be two vectors in $L$ and let $\alpha \in \mathbb{R}$ be a scalar. Then
$$\alpha w_1+w_2=\alpha \beta_1 v+\beta_2 v=\left(\alpha \beta_1+\beta_2\right) v,$$
and so $L$ is closed under addition and scalar multiplication.
Also, notice that $L$ is nonempty. Thus, by Theorem $2.5 .3, L$ is a subspace of $\left(\mathbb{R}^n,+, \cdot\right)$.
As we explore higher dimensions, we find that planes through the origin (and in fact any geometric space described by the solutions of a homogeneous system of linear equations) are also vector spaces. We define a plane as the set of all linear combinations of two (not colinear) vectors in $\mathbb{R}^3$. We leave it as Exercise 10 to write the proof of this fact as it follows, closely, the proof of Theorem 2.5.16.

## 数学代写|线性代数代写linear algebra代考|Building New Subspaces

In this section, we investigate the question, “If we start with two subspaces of the same vector space, in what ways can we combine the vectors to obtain another subspace?” Our investigation will lead us to some observations that will simplify some previous examples and give us new tools for proving that subsets are subspaces. We first consider intersections and unions.

Let $S$ and $T$ be sets.

• The intersection of $S$ and $T$, written $S \cap T$, is the set containing all elements that are in both $S$ and $T$.
• The union of $S$ and $T$, written $S \cup T$, is the set containing all elements that are in either $S$ or $T$ (or both).
The intersection of two subspaces is a also a subspace.

Proof. Let $W_1$ and $W_2$ be subspaces of $(V,+, \cdot)$. We will show that the intersection of $W_1$ and $W_2$ is nonempty and closed under scalar multiplication and vector addition. To show that $W_1 \cap W_2$ is nonempty, we notice that since both $W_1$ and $W_2$ contain the zero vector, so does $W_1 \cap W_2$.

Now, let $u$ and $v$ be elements of $W_1 \cap W_2$ and let $\alpha$ be a scalar. Since $W_1$ and $W_2$ are closed under addition and scalar multiplication, we know that $\alpha \cdot u+v$ is also in both $W_1$ and $W_2$. That is, $\alpha \cdot u+v$ is in $W_1 \cap W_2$, so by Corollary $2.5 .6 W_1 \cap W_2$ is closed under addition and scalar multiplication.
Thus, by Corollary $2.5 .4, W_1 \cap W_2$ is a subspace of $(V,+, \cdot)$.
An important example involves solutions to homogeneous equations, which we first considered in Example 2.4.12.

Example 2.5.19 The solution set of a single homogeneous equation in $n$ variables is a subspace of $\mathbb{R}^n$ (see Example 2.4.12). By Theorem 2.5.18, the intersection of the solution sets of any $k$ homogeneous equations in $n$ variables is also subspace of $\mathbb{R}^n$.

In other words, if a system of linear equations consists only of homogeneous equations, then the set of solutions forms a subspace of $\mathbb{R}^n$. This is such an important result that we promote it from example to theorem.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Subspaces of Rn

$L=\beta v \mid \beta \in \mathbb{R}$. 我们展示 $L$ 是使用推论 $2.5 .6$ 中给出 的属性的子空间。让 $w_1=\beta_1 v$ 和 $w_2=\beta_2 v$ 是两个向 量 $L$ 然后让 $\alpha \in \mathbb{R}$ 是一个标量。然后
$$\alpha w_1+w_2=\alpha \beta_1 v+\beta_2 v=\left(\alpha \beta_1+\beta_2\right) v$$

## 数学代写|线性代数代写linear algebra代考|Building New Subspaces

• 的交集 $S$ 和 $T$ ，写 $S \cap T$ ，是包含所有元素的集合 $S$ 和 $T$.
• 的联盟 $S$ 和 $T$, 写 $S \cup T$, 是包含所有元素的集合 $S$ 要么 $T$ (或两者)。 两个子空间的交集也是一个子空间。
证明。让 $W_1$ 和 $W_2$ 是子空间 $(V,+, \cdot)$. 我们将证明 $W_1$ 和 $W_2$ 在标量乘法和向量加法下是非空和封闭的。为了 表明 $W_1 \cap W_2$ 是非空的，我们注意到因为两者 $W_1$ 和 $W_2$ 包含零向量，所以 $W_1 \cap W_2$.
现在，让 $u$ 和 $v$ 成为元素 $W_1 \cap W_2$ 然后让 $\alpha$ 是一个标 量。自从 $W_1$ 和 $W_2$ 在加法和标量乘法下是封闭的，我们 知道 $\alpha \cdot u+v$ 也在两者之中 $W_1$ 和 $W_2$. 那是， $\alpha \cdot u+v$ 在 $W_1 \cap W_2$, 所以由推论 $2.5 .6 W_1 \cap W_2$ 在 加法和标量乘法下是封闭的。
因此，通过推论 $2.5 .4, W_1 \cap W_2$ 是一个子空间 $(V,+, \cdot)$.
一个重要的例子涉及齐次方程的解，我们在例 2.4.12 中 首先考虑了这一点。
例 2.5.19 单齐次方程的解集 $n$ 变量是的子空间 $\mathbb{R}^n$ （参见 示例 2.4.12）。根据定理 2.5.18，任意解集的交集 $k$ 中 的齐次方程 $n$ 变量也是子空间 $\mathbb{R}^n$.
换句话说，如果线性方程组仅由齐次方程组成，则解集 形成一个子空间 $\mathbb{R}^n$. 这是一个如此重要的结果，我们将 它从例子提升到定理。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。