# 数学代写|线性代数代写linear algebra代考|MATHS1011

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

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|The Contrast Between Nul A and Col A

It is natural to wonder how the null space and column space of a matrix are related. In fact, the two spaces are quite dissimilar, as Examples $5-7$ will show. Nevertheless, a surprising connection between the null space and column space will emerge in Section 4.6, after more theory is available.
EXAMPLE 5 Let
$$A=\left[\begin{array}{rrrr} 2 & 4 & -2 & 1 \ -2 & -5 & 7 & 3 \ 3 & 7 & -8 & 6 \end{array}\right]$$
a. If the column space of $A$ is a subspace of $\mathbb{R}^k$, what is $k$ ?
b. If the null space of $A$ is a subspace of $\mathrm{R}^k$, what is $k$ ?
SOLUTION
a. The columns of $A$ each have three entries, so $\operatorname{Col} A$ is a subspace of $\mathbb{R}^k$, where $k=3$.
b. A vector $\mathbf{x}$ such that $A \mathbf{x}$ is defined must have four entries, so $\mathrm{Nul} A$ is a subspace of $\mathbb{R}^k$, where $k=4$.

When a matrix is not square, as in Example 5, the vectors in $\mathrm{Nul} A$ and $\operatorname{Col} A$ live in entirely different “universes.” For example, no linear combination of vectors in $\mathbb{R}^3$ can produce a vector in $\mathbb{R}^4$. When $A$ is square, $\operatorname{Nul} A$ and $\operatorname{Col} A$ do have the zero vector in common, and in special cases it is possible that some nonzero vectors belong to both $\operatorname{Nul} A$ and $\operatorname{Col} A$.

## 数学代写|线性代数代写linear algebra代考|LINEARLY INDEPENDENT SETS; BASES

In this section we identify and study the subsets that span a vector space $V$ or a subspace $H$ as “efficiently” as possible. The key idea is that of linear independence, defined as in $\mathbb{R}^n$.

An indexed set of vectors $\left{\mathbf{v}_1, \ldots, \mathbf{v}_p\right}$ in $V$ is said to be linearly independent if the vector equation
$$c_1 \mathbf{v}_1+c_2 \mathbf{v}_2+\cdots+c_p \mathbf{v}_p=\mathbf{0}$$
has only the trivial solution, $c_1=0, \ldots, c_p=0^1$
The set $\left{\mathbf{v}_1, \ldots, \mathbf{v}_p\right}$ is said to be linearly dependent if $(1)$ has a nontrivial solution, that is, if there are some weights, $c_1, \ldots, c_p$, not all zero, such that (1) holds. In such a case, (1) is called a linear dependence relation among $\mathbf{v}_1, \ldots, \mathbf{v}_p$.

Just as in $\mathbb{R}^n$, a set containing a single vector $\mathbf{v}$ is linearly independent if and only if $\mathbf{v} \neq \mathbf{0}$. Also, a set of two vectors is linearly dependent if and only if one of the vectors is a multiple of the other. And any set containing the zero vector is linearly dependent. The following theorem has the same proof as Theorem 7 in Section 1.7.
An indexed set $\left{\mathbf{v}1, \ldots, \mathbf{v}_p\right}$ of two or more vectors, with $\mathbf{v}_1 \neq \mathbf{0}$, is linearly dependent if and only if some $\mathbf{v}_j$ (with $j>1$ ) is a linear combination of the preceding vectors, $\mathbf{v}_1, \ldots, \mathbf{v}{j-1}$.
The main difference between linear dependence in $\mathbb{R}^n$ and in a general vector space is that when the vectors are not $n$-tuples, the homogeneous equation (1) usually cannot be written as a system of $n$ linear equations. That is, the vectors cannot be made into the columns of a matrix $A$ in order to study the equation $A \mathbf{x}=\mathbf{0}$. We must rely instead on the definition of linear dependence and on Theorem 4 .

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|The Contrast Between Nul A and Col A

$$A=\left[\begin{array}{llllllllllll} 2 & 4 & -2 & 1 & -2 & -5 & 7 & 3 & 3 & 7 & -8 & 6 \end{array}\right]$$

b. 如果零空间 $A$ 是一个子空间 $\mathrm{R}^k$ ，什么是 $k$ ?

b. 向量 $\mathbf{x}$ 文样 $A \mathbf{x}$ 被定义必须有四个条目，所以Nul $A$ 是 一个子空间 $\mathbb{R}^k$ ， 在哪里 $k=4$.

$\mathrm{Nul} A$ 和Col $A$ 确实有共同的零向量，在特殊情况下，

## 数学代写|线性代数代写linear algebra代考|LINEARLY INDEPENDENT SETS; BASES

Ueft{\mathbf{v}_1, Vdots, Imathbf{v}_plright $}$ 在 $V$ 据说是线生无关的，如果向量方程
$$c_1 \mathbf{v}_1+c_2 \mathbf{v}_2+\cdots+c_p \mathbf{v}_p=\mathbf{0}$$

$\mathbf{v}_1, \ldots, \mathbf{v} j-1$

## 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。

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)