## 数学代写|线性代数代写linear algebra代考|MATH250

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## 数学代写|线性代数代写linear algebra代考|Transformation matrix for non-standard bases

Why use non-standard bases?
Examining a vector in a different basis (axes) may bring out structure related to that basis, which is hidden in the standard representation. It may be a relevant and useful structure. For example, we used to measure the motion of the planets in a basis (axes) with the earth at the centre. Then we discovered that putting the sun at the centre made life simpler – orbits were measured against a basis with the sun at the focus.

For some motions, such as projectiles, our standard basis ( $x y$ axes) may be the most suitable, but for studying other kinds of motions, such as orbits, a polar basis $(r, \theta)$ may work better.

If we use latitudes and longitudes to work out a map then we have been effectively using spherical polar coordinates $(r, \theta, \varphi)$ rather than our standard $x y z$ axes.

Another example is trying to find the forces on an aeroplane as shown in Fig. 5.29. The components parallel and perpendicular to the aeroplane are a lot more useful than the horizontal and vertical components.

In computer games and 3D design software we often want to rotate our $x y z$ axes (basis) to obtain new axes (basis) which are a lot more useful. (See question 7 of Exercises 5.5.)
In crystal structures, we need to use a basis which gives a cleaner set of coordinates called Miller indices. The Miller indices are coordinates used to specify direction and planes in a crystal or lattice. A vector from the origin to the lattice point is normally written in appropriate basis (axes) vectors and then the coordinates are given by the Miller indices.

Many problems in physics can be simplified due to their symmetrical properties if the right basis (axes) is chosen. Choosing a basis (axes) wisely can greatly reduce the amount of arithmetic you have to do.

## 数学代写|线性代数代写linear algebra代考|Composition of linear transformations (mappings)

What do you think the term onto transformation means?
An illustration of an onto transformation is shown in Fig. 5.24.

An onto transformation is when all the information carried over by $T$ fills the whole arrival vector space $W$.
How can we write this in mathematical terms?
Definition (5.18). Let $T: V \rightarrow W$ be a linear transform. The transform $T$ is onto $\Leftrightarrow$ for every $\mathbf{w}$ in the arrival vector space $W$ there exists at least one $\mathbf{v}$ in the start vector space $V$ such that
$$\mathbf{w}=T(\mathbf{v})$$
In other words $T: V \rightarrow W$ is an onto transformation $\Leftrightarrow \operatorname{range}(T)=W$. This means the arriving vectors of $T$ fill all of $W$. We can write this as a proposition:
Proposition (5.19). A linear transformation $T: V \rightarrow W$ is onto $\Leftrightarrow \operatorname{range}(T)=W$.
Proof – Exercises 5.4.

In other mathematical literature, or your lecture notes, you might find the term surjective to mean onto. We will use onto.

Remember that linear transforms are functions and you should be familiar with the concept of a function.
What does composition mean?
Composition means making something by combining parts.
What do you think composition of linear transformation means?
It is the linear transformation created by putting together two or more linear transformations.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Composition of linear transformations (mappings)

$$\mathbf{w}=T(\mathbf{v})$$

## 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。

## 数学代写|线性代数代写linear algebra代考|Definition of inner product

statistics-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。

## 数学代写|线性代数代写linear algebra代考|Definition of inner product

How did we define the inner or dot product in chapter 2?
Let $\mathbf{u}=\left(\begin{array}{c}u_1 \ \vdots \ u_n\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}v_1 \ \vdots \ v_n\end{array}\right)$ be vectors in $\mathbb{R}^n$ then the inner product of $\mathbf{u}$ and $\mathbf{v}$ denoted by $\mathbf{u} \cdot \mathbf{v}$ is
$$\mathbf{u} \cdot \mathbf{v}=\mathbf{u}^T \mathbf{v}=u_1 v_1+u_2 v_2+u_3 v_3+\cdots+u_n v_n$$
Remember, the answer was a scalar not a vector. This inner product was named the dot product (also called the scalar product) in $\mathbb{R}^n$. This is the usual (or standard) inner product in $\mathbb{R}^n$ but there are many other types of inner products in $\mathbb{R}^n$.

For the general vector space, the inner product is denoted by $\langle\mathbf{u}, \mathbf{v}\rangle$ rather than $\mathbf{u} \cdot \mathbf{v}$. For the general vector space, the definition of inner product is based on Proposition (2.6) of chapter 2 and is given by:

Definition (4.1). An inner product on a real vector space $V$ is an operation which assigns to each pair of vectors, $\mathbf{u}$ and $\mathbf{v}$, a unique real number $\langle\mathbf{u}, \mathbf{v}\rangle$ which satisfies the following axioms for all vectors $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ in $V$ and all scalars $k$.
(i) $\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle \quad$ [commutative law]
(ii) $\langle\mathbf{u}+\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{w}\rangle+\langle\mathbf{v}, \mathbf{w}\rangle \quad$ [distributive law]
(iii) $\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle \quad$ [taking out the scalar $k$ ]
(iv) $\langle\mathbf{u}, \mathbf{u}\rangle \geq 0$ and we have $\langle\mathbf{u}, \mathbf{u}\rangle=0 \Leftrightarrow \mathbf{u}=\mathbf{O}$ [Means the inner product between the same vectors is zero or positive.]

A real vector space which satisfies these axioms is called a real inner product space. Note that evaluating $\langle$,$\rangle gives a real number (scalar) not a vector. Next we give some examples$ of inner product spaces.

## 数学代写|线性代数代写linear algebra代考|Properties of inner products

Proposition (4.2). Let $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ be vectors in a real inner product space $V$ and $k$ be any real scalar. We have the following properties of inner products:
(i) $\langle\mathbf{u}, \mathbf{O}\rangle=\langle\mathbf{O}, \mathbf{v}\rangle=0$
(ii) $\langle\mathbf{u}, k \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle$
(iii) $\langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle$
How do we prove these properties?
We use the axioms of inner products stated in Definition (4.1).
Proof of (i).
We can write the zero vector as $0(\mathbf{O})$ because $0(\mathbf{O})=\mathbf{O}$. Using the axioms of definition (4.1) we have
\begin{aligned} \langle\mathbf{u}, \mathbf{O}\rangle & =\langle\mathbf{u}, 0(\mathbf{O})\rangle & & \ & =\langle 0(\mathbf{O}), \mathbf{u}\rangle & & {[\text { by part (i) of (4.1) which is }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \ & =0\langle\mathbf{O}, \mathbf{u}\rangle & & {[\text { by part (iii) of (4.1) which is }\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle] } \ & =0 & & \end{aligned}
Similarly $\langle\mathbf{O}, \mathbf{v}\rangle=0$.

Proof of (ii).
The inner product is commutative, $\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle$, which means we can switch the vectors around. We have
\begin{aligned} \langle\mathbf{u}, k \mathbf{v}\rangle & =\langle k \mathbf{v}, \mathbf{u}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \ & =k\langle\mathbf{v}, \mathbf{u}\rangle & & {[\text { by part (iii) of (4.1) which is }\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle] } \ & =k\langle\mathbf{u}, \mathbf{v}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \end{aligned}
Proof of (iii).
We have
\begin{aligned} \langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle & =\langle\mathbf{v}+\mathbf{w}, \mathbf{u}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \ & =\langle\mathbf{v}, \mathbf{u}\rangle+\langle\mathbf{w}, \mathbf{u}\rangle & & {\left[\begin{array}{c} \text { by part (ii) of }(4.1) \text { which is } \ \langle\mathbf{v}+\mathbf{w}, \mathbf{u}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle+\langle\mathbf{w}, \mathbf{u}\rangle \end{array}\right] } \ & =\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \end{aligned}

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Definition of inner product

$$\mathbf{u} \cdot \mathbf{v}=\mathbf{u}^T \mathbf{v}=u_1 v_1+u_2 v_2+u_3 v_3+\cdots+u_n v_n$$

(i) $\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle \quad$[交换律]
(ii) $\langle\mathbf{u}+\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{w}\rangle+\langle\mathbf{v}, \mathbf{w}\rangle \quad$[分配律]
(iii) $\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle \quad$[取出标量$k$]
(iv) $\langle\mathbf{u}, \mathbf{u}\rangle \geq 0$，我们有$\langle\mathbf{u}, \mathbf{u}\rangle=0 \Leftrightarrow \mathbf{u}=\mathbf{O}$[意味着相同向量之间的内积为零或正。]

## 数学代写|线性代数代写linear algebra代考|Properties of inner products

(i) $\langle\mathbf{u}, \mathbf{O}\rangle=\langle\mathbf{O}, \mathbf{v}\rangle=0$
(ii) $\langle\mathbf{u}, k \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle$
(三)$\langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle$

\begin{aligned} \langle\mathbf{u}, \mathbf{O}\rangle & =\langle\mathbf{u}, 0(\mathbf{O})\rangle & & \ & =\langle 0(\mathbf{O}), \mathbf{u}\rangle & & {[\text { by part (i) of (4.1) which is }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \ & =0\langle\mathbf{O}, \mathbf{u}\rangle & & {[\text { by part (iii) of (4.1) which is }\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle] } \ & =0 & & \end{aligned}

\begin{aligned} \langle\mathbf{u}, k \mathbf{v}\rangle & =\langle k \mathbf{v}, \mathbf{u}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \ & =k\langle\mathbf{v}, \mathbf{u}\rangle & & {[\text { by part (iii) of (4.1) which is }\langle k \mathbf{u}, \mathbf{v}\rangle=k\langle\mathbf{u}, \mathbf{v}\rangle] } \ & =k\langle\mathbf{u}, \mathbf{v}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \end{aligned}

\begin{aligned} \langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle & =\langle\mathbf{v}+\mathbf{w}, \mathbf{u}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \ & =\langle\mathbf{v}, \mathbf{u}\rangle+\langle\mathbf{w}, \mathbf{u}\rangle & & {\left[\begin{array}{c} \text { by part (ii) of }(4.1) \text { which is } \ \langle\mathbf{v}+\mathbf{w}, \mathbf{u}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle+\langle\mathbf{w}, \mathbf{u}\rangle \end{array}\right] } \ & =\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle & & {[\text { switching vectors }\langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle] } \end{aligned}

## 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。

## 数学代写|线性代数代写linear algebra代考|Rank of a matrix

statistics-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。

## 数学代写|线性代数代写linear algebra代考|Rank of a matrix

Consider the linear system $\mathbf{A x}=\mathbf{b}$. The augmented matrix $(\mathbf{A} \mid \mathbf{b})$ in row echelon form may produce zero rows, which means $0=0$, but these are not important in the solution of linear equations. It is the number (rank) of the non-zero rows in row echelon form which gives the solution of a linear system of equations. We will discover in the next section that the rank of matrix $\mathbf{A}$ and of the augmented matrix (A $\mid \mathbf{b}$ ) tell us if there are no, a unique or an infinite number of solutions.
The rank of a matrix gives the number of linearly independent rows in a matrix which means that all the rows that are linearly dependent are counted as one. For example, the following matrix has a rank of 1 :
\begin{aligned} & \mathrm{R}_1 \ & \mathrm{R}_2 \end{aligned}\left(\begin{array}{llll} 1 & 2 & 3 & 4 \ 2 & 4 & 6 & 8 \end{array}\right) \text { can be transformed to } \begin{gathered} \mathrm{R}_1 \ \mathrm{R}_2-2 \mathrm{R}_1 \end{gathered}\left(\begin{array}{llll} 1 & 2 & 3 & 4 \ 0 & 0 & 0 & 0 \end{array}\right)

The second row is double the first, so carrying out row operations results in a single independent row. The rank of a matrix measures the amount of important information represented by the matrix.

An application of linear algebra is the transfer of digital data which is normally stored as a matrix. In these fields it is important that data is transferred as fast and efficiently as possible without losing any of it. The concept of a rank is critical here because a matrix with a lower rank takes up less memory and time to be transferred. Low rank matrices are much more efficient in the sense that they are much less computationally expensive to deal with.
Computer graphics rely on matrices to generate and manipulate images. The rank of the matrix tells you the dimension of the image. For example, the matrix $\mathbf{A}=\left(\begin{array}{lll}1 & 1 & 1 \ 4 & 5 & 6 \ 2 & 2 & 2\end{array}\right)$ transforms a vector in 3D onto a 2D plane because matrix $\mathbf{A}$ does not have ‘full rank’ (the top and bottom rows are linearly dependent) as shown in Fig. 3.20.

## 数学代写|线性代数代写linear algebra代考|Can you recall what the term dimension of a vector space means?

It is the least number of axes needed to describe the vector space, or in other words, the number of vectors in the basis of a vector space.

The dimension of the row space of a matrix is called row rank and the dimension of the column space is called the column rank. Note that the row rank of a given matrix $\mathbf{A}$ is the number of non-zero row vectors in row echelon form matrix $\mathbf{R}$ because the non-zero rows form a basis for the row space.
$$\left.\mathbf{A}=\left(\begin{array}{c} \mathbf{a}1 \ \vdots \ \mathbf{a}_m \ \mathbf{a}{m+1} \ \vdots \end{array}\right) \quad \mathbf{R}=\left(\begin{array}{c} \mathbf{r}_1 \ \vdots \ \mathbf{r}_m \ \mathbf{O} \ \vdots \end{array}\right)\right} m \text { non-zero rows }$$

Row rank of matrix $\mathbf{A}=m$
The row rank of a matrix is called the rank of a matrix.
Definition (3.28). The rank of a matrix $\mathbf{A}$ is the row rank of $\mathbf{A}$.
The rank of matrix $\mathbf{A}$ is denoted by $\operatorname{rank}(\mathbf{A})$. Thus (3.28) says
$$\operatorname{rank}(\mathbf{A})=\operatorname{row} \operatorname{rank} \text { of } \mathbf{A}$$

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Rank of a matrix

\begin{aligned} & \mathrm{R}_1 \ & \mathrm{R}_2 \end{aligned}\left(\begin{array}{llll} 1 & 2 & 3 & 4 \ 2 & 4 & 6 & 8 \end{array}\right) \text { can be transformed to } \begin{gathered} \mathrm{R}_1 \ \mathrm{R}_2-2 \mathrm{R}_1 \end{gathered}\left(\begin{array}{llll} 1 & 2 & 3 & 4 \ 0 & 0 & 0 & 0 \end{array}\right)

## 数学代写|线性代数代写linear algebra代考|Can you recall what the term dimension of a vector space means?

$$\left.\mathbf{A}=\left(\begin{array}{c} \mathbf{a}1 \ \vdots \ \mathbf{a}_m \ \mathbf{a}{m+1} \ \vdots \end{array}\right) \quad \mathbf{R}=\left(\begin{array}{c} \mathbf{r}_1 \ \vdots \ \mathbf{r}_m \ \mathbf{O} \ \vdots \end{array}\right)\right} m \text { non-zero rows }$$

$$\operatorname{rank}(\mathbf{A})=\operatorname{row} \operatorname{rank} \text { of } \mathbf{A}$$

## 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。

## 数学代写|线性代数代写linear algebra代考|How do we prove these results?

statistics-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。

## 数学代写|线性代数代写linear algebra代考|How do we prove these results?

By using the definition of basis as described in the last section, that is:
Definition (3.15) A set of vectors is a basis for $V \Leftrightarrow$ it is linearly independent and spans $V$.
Proof of $(a)$
We are given that the vectors $\left{\mathbf{v}1, \mathbf{v}_2, \mathbf{v}_3, \ldots, \mathbf{v}_n\right}$ are linearly independent so we only need to show that these vectors also span $V$. Suppose there is a vector $\mathbf{w}$ in $V$ such that $$\mathbf{w}=k_1 \mathbf{v}_1+k_2 \mathbf{v}_2+k_3 \mathbf{v}_3+\cdots+k_n \mathbf{v}_n+k{n+1} \mathbf{v}{n+1}$$ where $k$ ‘s are scalars. [We are supposing that the given vectors do not span $V$ that is why we have added an extra vector $\left.\mathbf{v}{n+1}\right]$.
By the above Lemma (3.21) part (a):
In a $n$ dimension space, vectors $\left{\mathbf{v}1, \mathbf{v}_2, \cdots, \mathbf{v}_m\right}$ where $m>n$ are linearly dependent. The set of vectors $\left{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \ldots, \mathbf{v}_n, \mathbf{v}{n+1}\right}$ is linearly dependent so we can write the vector $\mathbf{v}{n+1}$ in terms of its preceding vectors, that is $$\mathbf{v}{n+1}=c_1 \mathbf{v}1+c_2 \mathbf{v}_2+c_3 \mathbf{v}_3+\cdots+c_n \mathbf{v}_n \text { where } c \text { ‘s are scalars }$$ Substituting this into $\left(^*\right)$ gives \begin{aligned} \mathbf{w} & =k_1 \mathbf{v}_1+k_2 \mathbf{v}_2+\cdots+k_n \mathbf{v}_n+k{n+1} \mathbf{v}{n+1} \ & =k_1 \mathbf{v}_1+k_2 \mathbf{v}_2+\cdots+k_n \mathbf{v}_n+k{n+1}\left(c_1 \mathbf{v}1+c_2 \mathbf{v}_2+\cdots+c_n \mathbf{v}_n\right) \ & =\left(k_1+k{n+1} c_1\right) \mathbf{v}1+\left(k_2+k{n+1} c_2\right) \mathbf{v}2+\cdots+\left(k_n+k{n+1} c_n\right) \mathbf{v}_n \end{aligned}
Thus the vector $\mathbf{w}$ can be written as a linear combination of the given linearly independent vectors $\left{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \ldots, \mathbf{v}_n\right}$. Thus $\left{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \cdots, \mathbf{v}_n\right}$ spans $V$, therefore it forms a basis for $V$.

## 数学代写|线性代数代写linear algebra代考|What do we need to show for this set of vectors to be a basis for V?

Required to prove that the set of vectors under consideration $\left{\mathbf{u}1, \mathbf{u}_2, \mathbf{u}_3, \ldots, \mathbf{u}_n\right}$ is linearly independent. Suppose $\left{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \ldots, \mathbf{u}_n\right}$ is linearly dependent then we can write the vector $\mathbf{u}_n$ in terms of its preceding vectors, that is $$\mathbf{u}_n=c_1 \mathbf{u}_1+c_2 \mathbf{u}_2+c_3 \mathbf{u}_3+\cdots+c{n-1} \mathbf{u}{n-1} \text { where } c \text { ‘s are scalars }$$ Substituting this into the above $(\dagger)$ gives \begin{aligned} \mathbf{w} & =k_1 \mathbf{u}_1+k_2 \mathbf{u}_2+\cdots+k{n-1} \mathbf{u}{n-1}+k_n \mathbf{u}_n \ & =k_1 \mathbf{u}_1+k_2 \mathbf{u}_2+\cdots+k{n-1} \mathbf{u}{n-1}+k_n\left(c_1 \mathbf{u}_1+c_2 \mathbf{u}_2+\cdots+c{n-1} \mathbf{u}{n-1}\right) \ & =\left(k_1+k_n c_1\right) \mathbf{u}_1+\left(k_2+k_n c_2\right) \mathbf{u}_2+\cdots+\left(k{n-1}+k_n c_{n-1}\right) \mathbf{u}{n-1} \end{aligned} This shows that the above $n-1$ vectors $\left{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \ldots, \mathbf{u}{n-1}\right}$ span $V$. This is impossible because, by the above Lemma (3.21), part (b):

If $n$ vectors $\left{\mathbf{u}_1, \mathbf{u}_2, \cdots, \mathbf{u}_n\right}$ span $V$ then $\left{\mathbf{u}_1, \mathbf{u}_2, \cdots, \mathbf{u}_m\right}$ where $m<n$ does not span $V$.
Fewer than $n$ vectors cannot span $V$. Thus $\left{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \ldots, \mathbf{u}_n\right}$ is linearly independent which means it is a basis for the given vector space $V$.
Theorem (3.22) says two things:

1. The maximum independent set for an $n$-dimensional vector space is $n$ vectors. If you add any more vectors then it becomes linearly dependent. The basis is a maximum linearly independent set.
2. The minimum spanning set for an $n$-dimensional vector space is $n$ vectors. If you remove any of the vectors of the spanning set then it no longer spans $V$. The basis is a minimum spanning set.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|What do we need to show for this set of vectors to be a basis for V?

$n$维向量空间的最大独立集是$n$个向量。如果你加上更多的向量，它就变成线性相关的了。基是一个极大线性无关的集合。

$n$维向量空间的最小生成集是$n$个向量。如果你移除了生成集的任何向量那么它就不再张成$V$。基是最小生成集。

## 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。

## 数学代写|线性代数代写linear algebra代考|Revision of linear combination

statistics-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。

## 数学代写|线性代数代写linear algebra代考|Revision of linear combination

Linear combination combines the two fundamental operations of linear algebra – vector addition and scalar multiplication.
In the last chapter we introduced linear combination in $\mathbb{R}^n$. For example, we had
$$\mathbf{u}=k_1 \mathbf{e}_1+k_2 \mathbf{e}_2+\cdots+k_n \mathbf{e}_n \quad(k \text { ‘s are scalars) }$$
which is a linear combination of the standard unit vectors $\mathbf{e}_1, \mathbf{e}_2, \ldots$ and $\mathbf{e}_n$.
Similarly for general vector spaces we define linear combination as:
Definition (3.6). Let $\mathbf{v}_1, \mathbf{v}_2, \ldots$ and $\mathbf{v}_n$ be vectors in a vector space. If a vector $\mathbf{x}$ can be expressed as
$$\mathbf{x}=k_1 \mathbf{v}_1+k_2 \mathbf{v}_2+k_3 \mathbf{v}_3+\cdots+k_n \mathbf{v}_n \text { (where } k \text { ‘s are scalars) }$$
then we say $\mathbf{x}$ is a linear combination of the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \ldots$ and $\mathbf{v}_n$.

## 数学代写|线性代数代写linear algebra代考|How do we prove this proposition?

Since it is a ‘ $\Leftrightarrow$ ‘ statement we need to prove it both ways.
Proof.
$(\Rightarrow)$. Let $S$ be a subspace of $V$, then $S$ is a vector space. If $\mathbf{u}$ and $\mathbf{v}$ are in $S$ then $k \mathbf{u}+c \mathbf{v}$ is also in $S$ because the vector space $S$ is closed under scalar multiplication and vector addition. $(\Leftarrow)$. Assume $k \mathbf{u}+c \mathbf{v}$ is in $S$.

Substituting $k=c=1$ into $k \mathbf{u}+c \mathbf{v}$ we have $\mathbf{u}+\mathbf{v}$ is also in $S$. Similarly for $c=0$ we have $k \mathbf{u}+c \mathbf{v}=k \mathbf{u}$ is in $S$.

Hence we have closure under vector addition and scalar multiplication. By Proposition (3.5): $S$ is subspace of $V \Leftrightarrow S$ is closed under vector addition and scalar multiplication.
We conclude that $S$ is a subspace.
Proposition (3.7) is another test for a subspace. Hence a subspace of a vector space $V$ is a non-empty subset $S$ of $V$, such that for all vectors $\mathbf{u}$ and $\mathbf{v}$ in $S$ and all scalars $k$ and $c$ we have $k \mathbf{u}+c \mathbf{v}$ is also in $S$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Revision of linear combination

$$\mathbf{u}=k_1 \mathbf{e}_1+k_2 \mathbf{e}_2+\cdots+k_n \mathbf{e}_n \quad(k \text { ‘s are scalars) }$$

$$\mathbf{x}=k_1 \mathbf{v}_1+k_2 \mathbf{v}_2+k_3 \mathbf{v}_3+\cdots+k_n \mathbf{v}_n \text { (where } k \text { ‘s are scalars) }$$

## 数学代写|线性代数代写linear algebra代考|How do we prove this proposition?

$(\Rightarrow)$。设$S$是$V$的一个子空间，那么$S$是一个向量空间。如果$\mathbf{u}$和$\mathbf{v}$在$S$中，那么$k \mathbf{u}+c \mathbf{v}$也在$S$中，因为向量空间$S$在标量乘法和向量加法下是封闭的。$(\Leftarrow)$。假设$k \mathbf{u}+c \mathbf{v}$在$S$中。

## 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。

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

statistics-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。

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

Let $V$ be a non-empty set of elements called vectors. We define two operations on the set $V$ – vector addition and scalar multiplication. Scalars are real numbers.

Let $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ be vectors in the set $V$. The set $V$ is called a vector space if it satisfies the following 10 axioms.

The vector addition $\mathbf{u}+\mathbf{v}$ is also in the vector space $V$. Generally in mathematics we say that we have closure under vector addition if this property holds.

Commutative law: $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$.

Associative law: $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$.

Neutral element. There is a vector called the zero vector in $V$ denoted by $\mathbf{O}$ which satisfies
$$\mathbf{u}+\mathbf{O}=\mathbf{u} \text { for every vector } \mathbf{u} \text { in } V$$

Additive inverse. For every vector $\mathbf{u}$ there is a vector $-\mathbf{u}$ (minus $\mathbf{u}$ ) which satisfies the following:
$$\mathbf{u}+(-\mathbf{u})=\mathbf{O}$$

Let $k$ be a real scalar then $k \mathbf{u}$ is also in $V$. We say that we have closure under scalar multiplication if this axiom is satisfied.

Associative law for scalar multiplication. Let $k$ and $c$ be real scalars then
$$k(c \mathbf{u})=(k c) \mathbf{u}$$

Distributive law for vectors. Let $k$ be a real scalar then
$$k(\mathbf{u}+\mathbf{v})=k \mathbf{u}+k \mathbf{v}$$

Distributive law for scalars. Let $k$ and $c$ be real scalars then
$$(k+c) \mathbf{u}=k \mathbf{u}+c \mathbf{u}$$

Identity element. For every vector $\mathbf{u}$ in $V$ we have
$$1 \mathbf{u}=\mathbf{u}$$
We say that if the elements of the set $V$ satisfy the above 10 axioms then $V$ is called a vector space and the elements are known as vectors. This might seem like a long list to digest, so don’t worry if it seems a little intimidating at this point. We will use these axioms frequently in the next few sections, and you will soon become familiar with them.

## 数学代写|线性代数代写linear algebra代考|Examples of vector spaces

Can you think of any examples of vector spaces?
The Euclidean spaces of the last chapter $-V=\mathbb{R}^2, \mathbb{R}^3, \ldots, \mathbb{R}^n$ – are all examples of vector spaces.
Are there any other examples of a vector space?
The set of matrices $M_{22}$ that are all matrices of size 2 by 2 where matrix addition and scalar multiplication is defined as in chapter 1 form their own vector space.
Let $\mathbf{u}=\left(\begin{array}{ll}a & b \ c & d\end{array}\right), \mathbf{v}=\left(\begin{array}{ll}e & f \ g & h\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{ll}i & j \ k & l\end{array}\right)$ be matrices in $M_{22}$.
What is the zero vector in $M_{22}$ ?
The zero vector is the zero matrix of size 2 by 2 which is given by $\left(\begin{array}{ll}0 & 0 \ 0 & 0\end{array}\right)=\mathbf{O}$.

The rules of matrix algebra established in chapter 1 ensure that all 10 axioms are satisfied, defining $M_{22}$ as an example of a vector space. You are asked to check this in Exercises 3.1.
We can show that the set $M_{23}$, which is the set of matrices of size 2 by 3 , also forms a vector space. You are asked to do this in Exercises 3.1.

There also exists vector space which is neither Euclidean space nor formed by a set of matrices. For example, the set of polynomials denoted $P(t)$ whose elements take the form:
$$\mathbf{p}(t)=c_0+c_1 t+c_2 t^2+\cdots+c_n t^n$$
where $c_0, c_1, c_2, \ldots$ and $c_n$ are real numbers called the coefficients, form a vector space. The following are examples of polynomials
$$\mathbf{p}(t)=t^2-1, \mathbf{q}(t)=1+2 t+7 t^2+12 t^3-3 t^4 \text { and } \mathbf{r}(t)=7$$
The degree of a polynomial is the highest index (power) which has a non-zero coefficient, that is the maximum $n$ for which $c_n \neq 0$.

# 线性代数代考

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

$$\mathbf{u}+\mathbf{O}=\mathbf{u} \text { for every vector } \mathbf{u} \text { in } V$$

$$\mathbf{u}+(-\mathbf{u})=\mathbf{O}$$

$$k(c \mathbf{u})=(k c) \mathbf{u}$$

$$k(\mathbf{u}+\mathbf{v})=k \mathbf{u}+k \mathbf{v}$$

$$(k+c) \mathbf{u}=k \mathbf{u}+c \mathbf{u}$$

$$1 \mathbf{u}=\mathbf{u}$$

## 数学代写|线性代数代写linear algebra代考|Examples of vector spaces

$M_{22}$中的零向量是什么?

$$\mathbf{p}(t)=c_0+c_1 t+c_2 t^2+\cdots+c_n t^n$$

$$\mathbf{p}(t)=t^2-1, \mathbf{q}(t)=1+2 t+7 t^2+12 t^3-3 t^4 \text { and } \mathbf{r}(t)=7$$

## 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。

## 数学代写|线性代数代写linear algebra代考|Standard unit vectors in R

statistics-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。

## 数学代写|线性代数代写linear algebra代考|Standard unit vectors in R

What does the term standard unit vector mean?
Recall from the last section that unit vectors are of length 1 , and standard unit vectors in $\mathbb{R}^n$ are column vectors with 1 in the $k$ th position of the vector $\mathbf{e}_k$ and zeros everywhere else (Fig. 2.26).

Why are these standard unit vectors important?
Because we can write any vector $\mathbf{u}$ of $\mathbb{R}^n$ in terms of scalars and standard unit vectors as we showed in Exercises 1.3, question 14. We proved the following important result:
Proposition (2.17). Let $\mathbf{u}=\left(x_1 \cdots x_k \cdots x_n\right)^T$ be any vector in $\mathbb{R}^n$ then
$$\mathbf{u}=\underbrace{x_1}{\text {scalar unit vector }} \underbrace{\mathbf{e}_1}{\text {scalar unit vector }} \underbrace{\mathbf{e}2}{\text {e }}+\cdots+\underbrace{x_k}{\text {scalar unit vector }} \mathbf{e}_k+\cdots+\underbrace{x_n}{\text {scalar unit vector }} \underbrace{\mathbf{e}^2}_{\mathbf{e}_n}$$
The position of vector $\mathbf{u}$ can be described (uniquely) by these scalars and unit vectors $\mathbf{e}_1, \mathbf{e}_2, \ldots$ and $\mathbf{e}_n$.
For example, the vector $\mathbf{u}=\left(\begin{array}{l}2 \ 3\end{array}\right)$ in $\mathbb{R}^2$ can be written as
$\left(\begin{array}{l}2 \ 3\end{array}\right)=2\left(\begin{array}{l}1 \ 0\end{array}\right)+3\left(\begin{array}{l}0 \ 1\end{array}\right)=2 \mathbf{e}_1+3 \mathbf{e}_2 \quad$ [In this case the scalars $x_1=2$ and $x_2=3$.]
Note that the scalars $x_1=2$ and $x_2=3$ are the coordinates of the vector $\mathbf{u}$.

This representation
$$\mathbf{u}=\left(\begin{array}{c} x_1 \ \vdots \ x_n \end{array}\right)=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_n \mathbf{e}_n$$
is a linear combination of the scalars and standard unit vectors $\mathbf{e}_1, \mathbf{e}_2, \ldots$ and $\mathbf{e}_n$. We can write this $\mathbf{u}=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_n \mathbf{e}_n$ in matrix form as
$$\mathbf{u}=\left(\begin{array}{lllll} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 & \cdots & \mathbf{e}_n \end{array}\right)\left(\begin{array}{c} x_1 \ \vdots \ x_n \end{array}\right) \text { where } \mathbf{e}_1=\left(\begin{array}{c} 1 \ 0 \ \vdots \end{array}\right), \ldots, \mathbf{e}_n=\left(\begin{array}{c} 0 \ \vdots \ 1 \end{array}\right)$$
The matrix $\left(\begin{array}{lllll}\mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 & \cdots & \mathbf{e}_n\end{array}\right)=\mathbf{I}$ where $\mathbf{I}$ is the identity matrix.
Proposition (2.18). Let $\mathbf{u}$ be any vector in $\mathbb{R}^n$ then the linear combination
$$\mathbf{u}=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_k \mathbf{e}_k+\cdots+x_n \mathbf{e}_n$$
is unique.

## 数学代写|线性代数代写linear algebra代考|What does this mean?

The only solution to the linear combination $k_1 \mathbf{v}_1+k_2 \mathbf{v}_2+\cdots+k_n \mathbf{v}_n=\mathbf{O}$ occurs when all the scalars $k_1, k_2, k_3, \ldots$ and $k_n$ are equal to zero. In other words you cannot make any one of the vectors $\mathbf{v}_j$, say, by a linear combination of the others.

We can write the linear combination $k_1 \mathbf{v}_1+k_2 \mathbf{v}_2+k_3 \mathbf{v}_3+\cdots+k_n \mathbf{v}_n=\mathbf{O}$ in matrix form as
$$\left(\begin{array}{llll} \mathbf{v}_1 & \mathbf{v}_2 & \cdots & \mathbf{v}_n \end{array}\right)\left(\begin{array}{c} k_1 \ \vdots \ k_n \end{array}\right)=\left(\begin{array}{c} 0 \ \vdots \ 0 \end{array}\right)$$
The first column of the matrix $\left(\begin{array}{llll}\mathbf{v}_1 & \mathbf{v}_2 & \cdots & \mathbf{v}_n\end{array}\right)$ is given by the entries in $\mathbf{v}_1$, the second column is given by the entries in $\mathbf{v}_2$ and the $n$th column by entries in $\mathbf{v}_n$.

The standard unit vectors are not the only vectors in $\mathbb{R}^n$ which are linearly independent. In the following example, we show another set of linearly independent vectors.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Standard unit vectors in R

$$\mathbf{u}=\underbrace{x_1}{\text {scalar unit vector }} \underbrace{\mathbf{e}1}{\text {scalar unit vector }} \underbrace{\mathbf{e}2}{\text {e }}+\cdots+\underbrace{x_k}{\text {scalar unit vector }} \mathbf{e}_k+\cdots+\underbrace{x_n}{\text {scalar unit vector }} \underbrace{\mathbf{e}^2}{\mathbf{e}_n}$$

$\left(\begin{array}{l}2 \ 3\end{array}\right)=2\left(\begin{array}{l}1 \ 0\end{array}\right)+3\left(\begin{array}{l}0 \ 1\end{array}\right)=2 \mathbf{e}_1+3 \mathbf{e}_2 \quad$[在本例中是标量$x_1=2$和$x_2=3$。]

$$\mathbf{u}=\left(\begin{array}{c} x_1 \ \vdots \ x_n \end{array}\right)=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_n \mathbf{e}_n$$

$$\mathbf{u}=\left(\begin{array}{lllll} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 & \cdots & \mathbf{e}_n \end{array}\right)\left(\begin{array}{c} x_1 \ \vdots \ x_n \end{array}\right) \text { where } \mathbf{e}_1=\left(\begin{array}{c} 1 \ 0 \ \vdots \end{array}\right), \ldots, \mathbf{e}_n=\left(\begin{array}{c} 0 \ \vdots \ 1 \end{array}\right)$$

$$\mathbf{u}=x_1 \mathbf{e}_1+x_2 \mathbf{e}_2+\cdots+x_k \mathbf{e}_k+\cdots+x_n \mathbf{e}_n$$

## 数学代写|线性代数代写linear algebra代考|What does this mean?

$$\left(\begin{array}{llll} \mathbf{v}_1 & \mathbf{v}_2 & \cdots & \mathbf{v}_n \end{array}\right)\left(\begin{array}{c} k_1 \ \vdots \ k_n \end{array}\right)=\left(\begin{array}{c} 0 \ \vdots \ 0 \end{array}\right)$$

## 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。

## 数学代写|线性代数代写linear algebra代考|The norm or length of a vector

statistics-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。

## 数学代写|线性代数代写linear algebra代考|The norm or length of a vector

Let $\mathbf{u}$ be a vector in $\mathbb{R}^n$. The length or norm of a vector $\mathbf{u}$ is denoted by $|\mathbf{u}|$. We can use Pythagoras’ theorem to find a way to define the norm of a vector. Consider a vector $\mathbf{u}$ in $\mathbb{R}^2$ (Fig. 2.6):

The length or norm of a vector $\mathbf{u}=\left(\begin{array}{ll}x & y\end{array}\right)^T$ in $\mathbb{R}^2$ is given by:
$$|\mathbf{u}|=\sqrt{x^2+y^2}$$

The length or norm of a vector $\mathbf{v}=\left(\begin{array}{lll}x & y & z\end{array}\right)^T$ in $\mathbb{R}^3$ is given by
$$|\mathbf{v}|=\sqrt{x^2+y^2+z^2}$$
Pythagoras’ Theorem. Let $\mathbf{u}$ be a vector in $\mathbb{R}^n$ then the length of $\mathbf{u}=\left(\begin{array}{c}u_1 \ \vdots \ u_n\end{array}\right)$ is given by:
$$|\mathbf{u}|=\sqrt{\left(u_1\right)^2+\left(u_2\right)^2+\left(u_3\right)^2+\cdots+\left(u_n\right)^2}$$
The norm of a vector $\mathbf{u}$ is a real number which gives the length of the vector $\mathbf{u}$.

## 数学代写|线性代数代写linear algebra代考|Properties of the norm of a vector

For a scalar $k$ we define the modulus of $k$ denoted $|k|$ as
$$|k|=\sqrt{k^2}$$
Proposition (2.10). Let $\mathbf{u}$ be a vector in $\mathbb{R}^n$ and $k$ be a real scalar. We have the following:
(i) $|\mathbf{u}| \geq 0$ [positive] and $|\mathbf{u}|=0 \Leftrightarrow \mathbf{u}=\mathbf{O}$.
(ii) $|k \mathbf{u}|=|k||\mathbf{u}|$

Proof.
Let $\mathbf{u}$ be a vector in $\mathbb{R}^n$; therefore we can write this as $\mathbf{u}=\left(u_1 \cdots u_n\right)^T$.
(i) Required to prove $|\mathbf{u}| \geq 0$. By Pythagoras’ theorem (2.7), we have the length of $\mathbf{u}$ :
$$|\mathbf{u}|=\sqrt{\left(u_1\right)^2+\left(u_2\right)^2+\left(u_3\right)^2+\cdots+\left(u_n\right)^2}$$
Since the square root is positive, $|\mathbf{u}| \geq 0$.
Next we prove the equality; that is $|\mathbf{u}|=0 \Leftrightarrow \mathbf{u}=\mathbf{O}$. We have $|\mathbf{u}|=0$, which means that
$$|\mathbf{u}|=\sqrt{\mathbf{u} \cdot \mathbf{u}}=0 \Leftrightarrow \mathbf{u} \cdot \mathbf{u}=0$$
By Proposition (2.6) part (iv), we have $\mathbf{u} \cdot \mathbf{u}=0 \Leftrightarrow \mathbf{u}=\mathbf{O}$. We have proven our equality.
(ii) Expanding the left hand side of $|k \mathbf{u}|=|k||\mathbf{u}|$ by applying definition; (2.8) $|\mathbf{v}|=\sqrt{\mathbf{v} \cdot \mathbf{v}}$ gives
\begin{aligned} |k \mathbf{u}|=\sqrt{k \mathbf{u} \cdot k \mathbf{u}} & =\sqrt{k^2(\mathbf{u} \cdot \mathbf{u})} \ & =\sqrt{k^2} \sqrt{\mathbf{u} \cdot \mathbf{u}} \ & =|k||\mathbf{u}| \end{aligned}
[because $k^2$ and $\mathbf{u} \cdot \mathbf{u}$ are real, so $\sqrt{a b}=\sqrt{a} \sqrt{b}$ ] [from above we have $\sqrt{k^2}=|k|$ ]

Normally, to obtain the length (norm) of a given vector $\mathbf{v}$ you will find it easier to determine $|\mathbf{v}|^2=\mathbf{v} \cdot \mathbf{v}$ and then take the square root of your result to find $|\mathbf{v}|$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|The norm or length of a vector

$\mathbb{R}^2$中向量$\mathbf{u}=\left(\begin{array}{ll}x & y\end{array}\right)^T$的长度或范数由下面给出:
$$|\mathbf{u}|=\sqrt{x^2+y^2}$$

$\mathbb{R}^3$中向量$\mathbf{v}=\left(\begin{array}{lll}x & y & z\end{array}\right)^T$的长度或范数由下式给出
$$|\mathbf{v}|=\sqrt{x^2+y^2+z^2}$$

$$|\mathbf{u}|=\sqrt{\left(u_1\right)^2+\left(u_2\right)^2+\left(u_3\right)^2+\cdots+\left(u_n\right)^2}$$

## 数学代写|线性代数代写linear algebra代考|Properties of the norm of a vector

$$|k|=\sqrt{k^2}$$

(i) $|\mathbf{u}| \geq 0$[正数]和$|\mathbf{u}|=0 \Leftrightarrow \mathbf{u}=\mathbf{O}$。
(ii) $|k \mathbf{u}|=|k||\mathbf{u}|$

(i)必须证明$|\mathbf{u}| \geq 0$。根据毕达哥拉斯定理(2.7)，我们得到$\mathbf{u}$的长度:
$$|\mathbf{u}|=\sqrt{\left(u_1\right)^2+\left(u_2\right)^2+\left(u_3\right)^2+\cdots+\left(u_n\right)^2}$$

$$|\mathbf{u}|=\sqrt{\mathbf{u} \cdot \mathbf{u}}=0 \Leftrightarrow \mathbf{u} \cdot \mathbf{u}=0$$

\begin{aligned} |k \mathbf{u}|=\sqrt{k \mathbf{u} \cdot k \mathbf{u}} & =\sqrt{k^2(\mathbf{u} \cdot \mathbf{u})} \ & =\sqrt{k^2} \sqrt{\mathbf{u} \cdot \mathbf{u}} \ & =|k||\mathbf{u}| \end{aligned}
[因为$k^2$和$\mathbf{u} \cdot \mathbf{u}$是实数，所以$\sqrt{a b}=\sqrt{a} \sqrt{b}$][从上面我们得到$\sqrt{k^2}=|k|$]

## 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。

## 数学代写|线性代数代写linear algebra代考|Non-invertible (singular) matrices

statistics-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。

## 数学代写|线性代数代写linear algebra代考|Non-invertible (singular) matrices

If the above approach of trying to convert (A|I) into the augmented matrix $\left(\mathbf{I} \mid \mathbf{A}^{-1}\right)$ cannot be achieved then the matrix $\mathbf{A}$ is non-invertible (singular). This means that the matrix A does not have an inverse. By Theorem (1.38), we know that the matrix $\mathbf{A}$ is invertible (has an inverse) $\Leftrightarrow$ the reduced row echelon form of $\mathbf{A}$ is the identity matrix I. Remember, the reduced row echelon form of an $n$ by $n$ matrix can only: (1) be an identity matrix or (2) have a row of zeros.
If we end up with a row of zeros then the given matrix is non-invertible.
Proposition (1.39). Let $\mathbf{A}$ be a square matrix and $\mathbf{R}$ be the reduced row echelon form of $\mathbf{A}$. Then $\mathbf{R}$ has at least one row of zeros $\Leftrightarrow \mathbf{A}$ is non-invertible (singular).
Proof.
See Exercises 1.8.
The proof of this result can be made a lot easier if we understand some mathematical logic. Generally to prove a statement of the type $P \Leftrightarrow Q$ we assume $P$ to be true and then deduce $Q$. Then we assume $Q$ to be true and deduce $P$.
However, in mathematical logic this can also be proven by showing:
$$(\operatorname{Not} P) \Leftrightarrow(\operatorname{Not} Q)$$
Means that $(\operatorname{Not} P) \Rightarrow(\operatorname{Not} Q)$ and $(\operatorname{Not} Q) \Rightarrow(\operatorname{Not} P)$. This is because statements $P \Leftrightarrow Q$ and $(\operatorname{Not} P) \Leftrightarrow($ Not $Q)$ are equivalent. See website for more details.
The following demonstrates this proposition.

## 数学代写|线性代数代写linear algebra代考|Applications to cryptography

Cryptography is the study of communication by stealth. It involves the coding and decoding of messages. This is a growing area of linear algebra applications because agencies such as the CIA use cryptography to encode and decode information.
One way to code a message is to use matrices.

For example, let $\mathbf{A}$ be an invertible matrix. The message is encrypted into a matrix $\mathbf{B}$ such that the matrix multiplication $\mathbf{A B}$ is a valid operation. Send the message generated by the matrix multiplication $\mathbf{A B}$. At the other end, they will need to know the inverse matrix $\mathbf{A}^{-1}$ in order to decode the message because $\mathbf{A}^{-1}(\mathbf{A B})=\mathbf{B}$. Remember that the matrix B contains the message.

A simple way of encoding messages is to represent each letter of the alphabet by its position in the alphabet and then add 3 to this. For example, we can create Table 1.3.

The final column represents space and we nominate this by a value of $27+3=30$.
To eliminate tedium from calculations, we use appropriate software to carry out the following example.

# 线性代数代考

## 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。