## 数学代写|线性代数代写linear algebra代考|MTH2106

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。

• 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|线性代数代写linear algebra代考|The Dimension of a Subspace

It can be shown that if a subspace $H$ has a basis of $p$ vectors, then every basis of $H$ must consist of exactly $p$ vectors. (See Exercises 27 and 28 .) Thus the following definition makes sense.
The dimension of a nonzero subspace $H$, denoted by $\operatorname{dim} H$, is the number of vectors in any basis for $H$. The dimension of the zero subspace ${0}$ is defined to be zero. ${ }^2$
The space $\mathbb{R}^n$ has dimension $n$. Every basis for $\mathbb{R}^n$ consists of $n$ vectors. A plane through 0 in $\mathbb{R}^3$ is two-dimensional, and a line through $\mathbf{0}$ is one-dimensional.

EXAMPLE 2 Recall that the null space of the matrix $A$ in Example 6 in Section $2.8$ had a basis of 3 vectors. So the dimension of $\operatorname{Nul} A$ in this case is 3 . Observe how each basis vector corresponds to a free variable in the equation $A \mathbf{x}=\mathbf{0}$. Our construction always produces a basis in this way. So, to find the dimension of $\mathrm{Nul} A$, simply identify and count the number of free variables in $A \mathbf{x}=\mathbf{0}$.
The rank of a matrix $A$, denoted by rank $A$, is the dimension of the column space of $A$.
Since the pivot columns of $A$ form a basis for $\operatorname{Col} A$, the rank of $A$ is just the number of pivot columns in $A$.

The row reduction in Example 3 reveals that there are two free variables in $A \mathbf{x}=\mathbf{0}$, because two of the five columns of $A$ are not pivot columns. (The nonpivot columns correspond to the free variables in $A \mathbf{x}=\mathbf{0}$.) Since the number of pivot columns plus the number of nonpivot columns is exactly the number of columns, the dimensions of Col $A$ and $\mathrm{Nul} A$ have the following useful connection. (See the Rank Theorem in Section $4.6$ for additional details.)
The Rank Theorem
If a matrix $A$ has $n$ columns, then $\operatorname{rank} A+\operatorname{dim} \operatorname{Nul} A=n$.
The following theorem is important for applications and will be needed in Chapters 5 and 6. The theorem (proved in Section 4.5) is certainly plausible, if you think of a $p$-dimensional subspace as isomorphic to $\mathbb{R}^p$. The Invertible Matrix Theorem shows that $p$ vectors in $\mathbb{R}^p$ are linearly independent if and only if they also span $\mathbb{R}^p$.

## 数学代写|线性代数代写linear algebra代考|Column Space and Null Space of a Matrix

Subspaces of $\mathbb{R}^n$ usually occur in applications and theory in one of two ways. In both cases, the subspace can be related to a matrix.
The column space of a matrix $A$ is the set $\operatorname{Col} A$ of all linear combinations of the columns of $A$.
If $A=\left[\begin{array}{lll}\mathbf{a}_1 & \cdots & \mathbf{a}_n\end{array}\right]$, with the columns in $\mathbb{R}^m$, then $\operatorname{Col} A$ is the same as Span $\left{\mathbf{a}_1, \ldots, \mathbf{a}_n\right}$. Example 4 shows that the column space of an $\boldsymbol{m} \times \boldsymbol{n}$ matrix is a subspace of $\mathbb{R}^m$. Note that $\operatorname{Col} A$ equals $\mathbb{R}^m$ only when the columns of $A$ span $\mathbb{R}^m$. Otherwise, $\operatorname{Col} A$ is only part of $\mathbb{R}^m$.

EXAMPLE 4 Let $A=\left[\begin{array}{rrr}1 & -3 & -4 \ -4 & 6 & -2 \ -3 & 7 & 6\end{array}\right]$ and $\mathbf{b}=\left[\begin{array}{r}3 \ 3 \ -4\end{array}\right]$. Determine whether $\mathbf{b}$ is in the column space of $A$.

SOLUTION The vector $\mathbf{b}$ is a linear combination of the columns of $A$ if and only if $\mathbf{b}$ can be written as $A \mathbf{x}$ for some $\mathbf{x}$, that is, if and only if the equation $A \mathbf{x}=\mathbf{b}$ has a solution. Row reducing the augmented matrix $\left[A \begin{array}{ll}A & \mathbf{b}\end{array}\right]$,
$$\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ -4 & 6 & -2 & 3 \ -3 & 7 & 6 & -4 \end{array}\right] \sim\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ 0 & -6 & -18 & 15 \ 0 & -2 & -6 & 5 \end{array}\right] \sim\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ 0 & -6 & -18 & 15 \ 0 & 0 & 0 & 0 \end{array}\right]$$
we conclude that $A \mathbf{x}=\mathbf{b}$ is consistent and $\mathbf{b}$ is in $\operatorname{Col} A$.

The solution of Example 4 shows that when a system of linear equations is written in the form $A \mathbf{x}=\mathbf{b}$, the column space of $A$ is the set of all $\mathbf{b}$ for which the system has a solution.
The null space of a matrix $A$ is the set $\mathrm{Nul} A$ of all solutions of the homogeneous equation $A \mathbf{x}=\mathbf{0}$
When $A$ has $n$ columns, the solutions of $A \mathbf{x}=\mathbf{0}$ belong to $\mathbb{R}^n$, and the null space of $A$ is a subset of $\mathbb{R}^n$. In fact, $\mathrm{Nul} A$ has the properties of a subspace of $\mathbb{R}^n$.
The null space of an $m \times n$ matrix $A$ is a subspace of $\mathbb{R}^n$. Equivalently, the set of all solutions of a system $A \mathbf{x}=\mathbf{0}$ of $m$ homogeneous linear equations in $n$ unknowns is a subspace of $\mathbb{R}^n$.
PROOF The zero vector is in $\operatorname{Nul} A$ (because $A 0=0$ ). To show that $\mathrm{Nul} A$ satisfies the other two properties required for a subspace, take any $\mathbf{u}$ and $\mathbf{v}$ in $\mathrm{Nul} A$. That is, suppose $A \mathbf{u}=\mathbf{0}$ and $A \mathbf{v}=\mathbf{0}$. Then, by a property of matrix multiplication,
$$A(\mathbf{u}+\mathbf{v})=A \mathbf{u}+A \mathbf{v}=\mathbf{0}+\mathbf{0}=\mathbf{0}$$
Thus $\mathbf{u}+\mathbf{v}$ satisfies $A \mathbf{x}=\mathbf{0}$, and so $\mathbf{u}+\mathbf{v}$ is in $\operatorname{Nul} A$. Also, for any scalar $c, A(c \mathbf{u})=$ $c(A \mathbf{u})=c(0)=\mathbf{0}$, which shows that $c \mathbf{u}$ is in $\mathrm{Nul} A$.

To test whether a given vector $\mathbf{v}$ is in $\operatorname{Nul} A$, just compute $A \mathbf{v}$ to see whether $A \mathbf{v}$ is the zero vector. Because $\mathrm{Nul} A$ is described by a condition that must be checked for each vector, we say that the null space is defined implicitly. In contrast, the column space is defined explicitly, because vectors in Col A can be constructed (by linear combinations) from the columns of $A$. To create an explicit description of $\mathrm{Nul} A$, solve the equation $A \mathbf{x}=\mathbf{0}$ and write the solution in parametric vector form. (See Example 6 , below.) ${ }^2$

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Column Space and Null Space of a Matrix

Veft{ $\left.\backslash m a t h b f{a} _1, \backslash d o t s, \backslash m a t h b f{a} _n \backslash r i g h t\right}$. 示例 4 显示了一个列空间 $\boldsymbol{m} \times \boldsymbol{n}$ 矩阵是一个子空间 $\mathbb{R}^m$. 注意 $\operatorname{Col} A$ 等于 $\mathbb{R}^m$ 只有当列 $A$ 跨度 $\mathbb{R}^m$. 否则， $\operatorname{Col} A$ 只是一部分 $\mathbb{R}^m$. $A$.

$$A(\mathbf{u}+\mathbf{v})=A \mathbf{u}+A \mathbf{v}=\mathbf{0}+\mathbf{0}=\mathbf{0}$$

## 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|MATH1051

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。

• 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|线性代数代写linear algebra代考|Perspective Projections

A three-dimensional object is represented on the two-dimensional computer screen by projecting the object onto a viewing plane. (We ignore other important steps, such as selecting the portion of the viewing plane to display on the screen.) For simplicity, let the $x y$-plane represent the computer screen, and imagine that the eye of a viewer is along the positive $z$-axis, at a point $(0,0, d)$. A perspective projection maps each point $(x, y, z)$ onto an image point $\left(x^, y^, 0\right)$ so that the two points and the eye position, called the center of projection, are on a line. See Figure 6(a).

The triangle in the $x z$-plane in Figure 6(a) is redrawn in part (b) showing the lengths of line segments. Similar triangles show that
$$\frac{x^}{d}=\frac{x}{d-z} \quad \text { and } \quad x^=\frac{d x}{d-z}=\frac{x}{1-z / d}$$
Similarly,
$$y^*=\frac{y}{1-z / d}$$
Using homogeneous coordinates, we can represent the perspective projection by a matrix, say, $P$. We want $(x, y, z, 1)$ to map into $\left(\frac{x}{1-z / d}, \frac{y}{1-z / d}, 0,1\right)$. Scaling these coordinates by $1-z / d$, we can also use $(x, y, 0,1-z / d)$ as homogeneous coordinates for the image. Now it is easy to display $P$. In fact,
$$P\left[\begin{array}{l} x \ y \ z \ 1 \end{array}\right]=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & -1 / d & 1 \end{array}\right]\left[\begin{array}{c} x \ y \ z \ 1 \end{array}\right]=\left[\begin{array}{c} x \ y \ 0 \ 1-z / d \end{array}\right]$$
EXAMPLE 8 Let $S$ be the box with vertices $(3,1,5),(5,1,5),(5,0,5),(3,0,5)$, $(3,1,4),(5,1,4),(5,0,4)$, and $(3,0,4)$. Find the image of $S$ under the perspective projection with center of projection at $(0,0,10)$.

## 数学代写|线性代数代写linear algebra代考|Column Space and Null Space of a Matrix

Subspaces of $\mathbb{R}^n$ usually occur in applications and theory in one of two ways. In both cases, the subspace can be related to a matrix.
The column space of a matrix $A$ is the set $\operatorname{Col} A$ of all linear combinations of the columns of $A$.
If $A=\left[\begin{array}{lll}\mathbf{a}_1 & \cdots & \mathbf{a}_n\end{array}\right]$, with the columns in $\mathbb{R}^m$, then $\operatorname{Col} A$ is the same as Span $\left{\mathbf{a}_1, \ldots, \mathbf{a}_n\right}$. Example 4 shows that the column space of an $\boldsymbol{m} \times \boldsymbol{n}$ matrix is a subspace of $\mathbb{R}^m$. Note that $\operatorname{Col} A$ equals $\mathbb{R}^m$ only when the columns of $A$ span $\mathbb{R}^m$. Otherwise, $\operatorname{Col} A$ is only part of $\mathbb{R}^m$.

EXAMPLE 4 Let $A=\left[\begin{array}{rrr}1 & -3 & -4 \ -4 & 6 & -2 \ -3 & 7 & 6\end{array}\right]$ and $\mathbf{b}=\left[\begin{array}{r}3 \ 3 \ -4\end{array}\right]$. Determine whether $\mathbf{b}$ is in the column space of $A$.

SOLUTION The vector $\mathbf{b}$ is a linear combination of the columns of $A$ if and only if $\mathbf{b}$ can be written as $A \mathbf{x}$ for some $\mathbf{x}$, that is, if and only if the equation $A \mathbf{x}=\mathbf{b}$ has a solution. Row reducing the augmented matrix $\left[A \begin{array}{ll}A & \mathbf{b}\end{array}\right]$,
$$\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ -4 & 6 & -2 & 3 \ -3 & 7 & 6 & -4 \end{array}\right] \sim\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ 0 & -6 & -18 & 15 \ 0 & -2 & -6 & 5 \end{array}\right] \sim\left[\begin{array}{rrrr} 1 & -3 & -4 & 3 \ 0 & -6 & -18 & 15 \ 0 & 0 & 0 & 0 \end{array}\right]$$
we conclude that $A \mathbf{x}=\mathbf{b}$ is consistent and $\mathbf{b}$ is in $\operatorname{Col} A$.

The solution of Example 4 shows that when a system of linear equations is written in the form $A \mathbf{x}=\mathbf{b}$, the column space of $A$ is the set of all $\mathbf{b}$ for which the system has a solution.
The null space of a matrix $A$ is the set $\mathrm{Nul} A$ of all solutions of the homogeneous equation $A \mathbf{x}=\mathbf{0}$
When $A$ has $n$ columns, the solutions of $A \mathbf{x}=\mathbf{0}$ belong to $\mathbb{R}^n$, and the null space of $A$ is a subset of $\mathbb{R}^n$. In fact, $\mathrm{Nul} A$ has the properties of a subspace of $\mathbb{R}^n$.
The null space of an $m \times n$ matrix $A$ is a subspace of $\mathbb{R}^n$. Equivalently, the set of all solutions of a system $A \mathbf{x}=\mathbf{0}$ of $m$ homogeneous linear equations in $n$ unknowns is a subspace of $\mathbb{R}^n$.
PROOF The zero vector is in $\operatorname{Nul} A$ (because $A 0=0$ ). To show that $\mathrm{Nul} A$ satisfies the other two properties required for a subspace, take any $\mathbf{u}$ and $\mathbf{v}$ in $\mathrm{Nul} A$. That is, suppose $A \mathbf{u}=\mathbf{0}$ and $A \mathbf{v}=\mathbf{0}$. Then, by a property of matrix multiplication,
$$A(\mathbf{u}+\mathbf{v})=A \mathbf{u}+A \mathbf{v}=\mathbf{0}+\mathbf{0}=\mathbf{0}$$
Thus $\mathbf{u}+\mathbf{v}$ satisfies $A \mathbf{x}=\mathbf{0}$, and so $\mathbf{u}+\mathbf{v}$ is in $\operatorname{Nul} A$. Also, for any scalar $c, A(c \mathbf{u})=$ $c(A \mathbf{u})=c(0)=\mathbf{0}$, which shows that $c \mathbf{u}$ is in $\mathrm{Nul} A$.

To test whether a given vector $\mathbf{v}$ is in $\operatorname{Nul} A$, just compute $A \mathbf{v}$ to see whether $A \mathbf{v}$ is the zero vector. Because $\mathrm{Nul} A$ is described by a condition that must be checked for each vector, we say that the null space is defined implicitly. In contrast, the column space is defined explicitly, because vectors in Col A can be constructed (by linear combinations) from the columns of $A$. To create an explicit description of $\mathrm{Nul} A$, solve the equation $A \mathbf{x}=\mathbf{0}$ and write the solution in parametric vector form. (See Example 6 , below.) ${ }^2$

# 线性代数代考

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

y^*=\frac{y}{1-z / d}
$$使用齐次坐标，我们可以用矩阵表示透视投影，比如说， P. 我们想要 (x, y, z, 1) 映射到 \left(\frac{x}{1-z / d}, \frac{y}{1-z / d}, 0,1\right). 缩放这些坐标 1-z / d ，我们也可以使用 (x, y, 0,1-z / d) 作为图像的齐次坐 标。现在很容易显示 P. 实际上，$$
P[x y z 1]=\left[\begin{array}{llllllllllllllll}
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 / d & 1
\end{array}\right]\left[\begin{array}{ll}
x y z & -1
\end{array}\right]=\left[\begin{array}{lll}
x & 0 & 1
\end{array}\right.
$$例 8 让 S 是有顶点的盒子 (3,1,5),(5,1,5),(5,0,5),(3,0,5) ，(3,1,4),(5,1,4),(5,0,4) ， 和 (3,0,4). 找到图像 S 在投影中心位于的透视投影下 (0,0,10). ## 数学代写|线性代数代写linear algebra代考|Column Space and Null Space of a Matrix 通过将物体投影到观察平面上，三维物体在二维计算机屏幕上呈现。（我们忽略其他重要步㡜，例如选择 要在屏幕上显示的视图平面部分。) 为简单起见，让 x y-plane 代表计算机屏幕，并想象观众的眼睛沿着 正面 z-轴，在一点 (0,0, d). 透视投影映射每个点 (x, y, z) 到图像点 \ V \mathrm{eft}\left(\mathrm{X}^{\wedge}, y^{\wedge} ， 0 \backslash r i g h t\right) \$$ 上，这样两 个点和眼睛位置 (称为投影中心) 在一条线上。见图 6(a)。

$\left\langle f r a c\left{x^{\wedge}\right}{d}=|f r a c{x}{d z} \backslash q u a d| t e x t{\right.$ 和 $\left.} \backslash q u a d x^{\wedge}=\right| f r a c{d x}{d z}=\backslash f r a c{x}{1-z / d}$

$$y^*=\frac{y}{1-z / d}$$

$$P[x y z 1]=\left[\begin{array}{llllllllllllllll} 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 / d & 1 \end{array}\right]\left[\begin{array}{ll} x y z & -1 \end{array}\right]=\left[\begin{array}{lll} x & 0 & 1 \end{array}\right.$$

## 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|MATH1014

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。

• 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|线性代数代写linear algebra代考|APPLICATIONS TO COMPUTER GRAPHICS

Computer graphics are images displayed or animated on a computer screen. Applications of computer graphics are widespread and growing rapidly. For instance, computeraided design (CAD) is an integral part of many engineering processes, such as the aircraft design process described in the chapter introduction. The entertainment industry has made the most spectacular use of computer graphics – from the special effects in Amazing Spider-Man 2 to PlayStation 4 and Xbox One.

Most interactive computer software for business and industry makes use of computer graphics in the screen displays and for other functions, such as graphical display of data, desktop publishing, and slide production for commercial and educational presentations. Consequently, anyone studying a computer language invariably spends time learning how to use at least two-dimensional (2D) graphics.

This section examines some of the basic mathematics used to manipulate and display graphical images such as a wire-frame model of an airplane. Such an image (or picture) consists of a number of points, connecting lines or curves, and information about how to fill in closed regions bounded by the lines and curves. Often, curved lines are approximated by short straight-line segments, and a figure is defined mathematically by a list of points.

Among the simplest 2D graphics symbols are letters used for labels on the screen. Some letters are stored as wire-frame objects; others that have curved portions are stored with additional mathematical formulas for the curves.

EXAMPLE 1 The capital letter $\mathrm{N}$ in Figure 1 is determined by eight points, or vertices. The coordinates of the points can be stored in a data matrix, $D$.
$x$-coordinate $\left[\begin{array}{cccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \ 0 & .5 & .5 & 6 & 6 & 5.5 & 5.5 & 0 \ 0 & 0 & 6.42 & 0 & 8 & 8 & 1.58 & 8\end{array}\right]=D$
In addition to $D$, it is necessary to specify which vertices are connected by lines, but we omit this detail.

The main reason graphical objects are described by collections of straight-line segments is that the standard transformations in computer graphics map line segments onto other line segments. (For instance, see Exercise 27 in Section 1.8.) Once the vertices that describe an object have been transformed, their images can be connected with the appropriate straight lines to produce the complete image of the original object.

## 数学代写|线性代数代写linear algebra代考|Homogeneous 3D Coordinates

By analogy with the $2 \mathrm{D}$ case, we say that $(x, y, z, 1)$ are homogeneous coordinates for the point $(x, y, z)$ in $\mathbb{R}^3$. In general, $(X, Y, Z, H)$ are homogeneous coordinates for $(x, y, z)$ if $H \neq 0$ and
$$x=\frac{X}{H}, \quad y=\frac{Y}{H}, \quad \text { and } \quad z=\frac{Z}{H}$$
Each nonzero scalar multiple of $(x, y, z, 1)$ gives a set of homogeneous coordinates for $(x, y, z)$. For instance, both $(10,-6,14,2)$ and $(-15,9,-21,-3)$ are homogeneous coordinates for $(5,-3,7)$.

The next example illustrates the transformations used in molecular modeling to move a drug into a protein molecule.
EXAMPLE 7 Give $4 \times 4$ matrices for the following transformations:
a. Rotation about the $y$-axis through an angle of $30^{\circ}$. (By convention, a positive angle is the counterclockwise direction when looking toward the origin from the positive half of the axis of rotation-in this case, the $y$-axis.)
b. Translation by the vector $\mathbf{p}=(-6,4,5)$.
SOLUTION
a. First, construct the $3 \times 3$ matrix for the rotation. The vector $\mathbf{e}_1$ rotates down toward the negative $z$-axis, stopping at $\left(\cos 30^{\circ}, 0,-\sin 30^{\circ}\right)=(\sqrt{3} / 2,0,-.5)$. The vector $\mathbf{e}_2$ on the $y$-axis does not move, but $\mathbf{e}_3$ on the $z$-axis rotates down toward the positive $x$-axis, stopping at $\left(\sin 30^{\circ}, 0, \cos 30^{\circ}\right)=(.5,0, \sqrt{3} / 2)$. See Figure 5. From Section $1.9$, the standard matrix for this rotation is
$$\left[\begin{array}{ccc} \sqrt{3} / 2 & 0 & .5 \ 0 & 1 & 0 \ -.5 & 0 & \sqrt{3} / 2 \end{array}\right]$$
So the rotation matrix for homogeneous coordinates is
$$A=\left[\begin{array}{cccc} \sqrt{3} / 2 & 0 & .5 & 0 \ 0 & 1 & 0 & 0 \ -.5 & 0 & \sqrt{3} / 2 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$
b. We want $(x, y, z, 1)$ to map to $(x-6, y+4, z+5,1)$. The matrix that does this is
$$\left[\begin{array}{rrrr} 1 & 0 & 0 & -6 \ 0 & 1 & 0 & 4 \ 0 & 0 & 1 & 5 \ 0 & 0 & 0 & 1 \end{array}\right]$$

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|APPLICATIONS TO COMPUTER GRAPHICS

$\left[\begin{array}{llllllllllllllllllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 0 & .5 & .5 & 6 & 6 & 5.5 & 5.5 & 0 & 0 & 0 & 6.42 & 0 & 8 & 8 & 1.58 & 8\end{array}\right]$ 此外 $D$ ，有必要指定哪些顶点由线连接，但我们省略了这个细节。

## 数学代写|线性代数代写linear algebra代考|Homogeneous 3D Coordinates

$$x=\frac{X}{H}, \quad y=\frac{Y}{H}, \quad \text { and } \quad z=\frac{Z}{H}$$

a。旋转关于 $y$-轴通过一个角度 $30^{\circ}$. (按照惯例，正角是从旋转轴的正半边看原点时的逆时针方向一一在 这种情况下， $y$-轴。)
$\mathrm{b}$ 。向量翻译 $\mathbf{p}=(-6,4,5)$.

$\left(\cos 30^{\circ}, 0,-\sin 30^{\circ}\right)=(\sqrt{3} / 2,0,-.5)$. 载体 $\mathbf{e}_2$ 在 $y$-轴不移动，但 $\mathbf{e}_3$ 在 $z$-axis 向下旋转到正 $x$ 轴，停在 $\left(\sin 30^{\circ}, 0, \cos 30^{\circ}\right)=(.5,0, \sqrt{3} / 2)$. 参见图 5。从部分 $1.9$ ，这个旋转的标准矩阵是

b. 我们想要 $(x, y, z, 1)$ 映射到 $(x-6, y+4, z+5,1)$. 这样做的矩阵是

## 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|MTH2106

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。

• 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|线性代数代写linear algebra代考|ROW REDUCTION AND ECHELON FORMS

This section refines the method of Section $1.1$ into a row reduction algorithm that will enable us to analyze any system of linear equations. ${ }^1$ By using only the first part of the algorithm, we will be able to answer the fundamental existence and uniqueness questions posed in Section 1.1.

The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system. So the first part of this section concerns an arbitrary rectangular matrix and begins by introducing two important classes of matrices that include the “triangular” matrices of Section 1.1. In the definitions that follow, a nonzero row or column in a matrix means a row or column that contains at least one nonzero entry; a leading entry of a row refers to the leftmost nonzero entry (in a nonzero row).

An echelon matrix (respectively, reduced echelon matrix) is one that is in echelon form (respectively, reduced echelon form). Property 2 says that the leading entries form an echelon (“steplike”) pattern that moves down and to the right through the matrix. Property 3 is a simple consequence of property 2 , but we include it for emphasis.
The “triangular” matrices of Section 1.1, such as
$$\left[\begin{array}{rrrc} 2 & -3 & 2 & 1 \ 0 & 1 & -4 & 8 \ 0 & 0 & 0 & 5 / 2 \end{array}\right] \text { and }\left[\begin{array}{lllr} 1 & 0 & 0 & 29 \ 0 & 1 & 0 & 16 \ 0 & 0 & 1 & 3 \end{array}\right]$$
are in echelon form. In fact, the second matrix is in reduced echelon form. Here are additional examples.

EXAMPLE 1 The following matrices are in echelon form. The leading entries ( $\boldsymbol{)}$ ) may have any nonzero value; the starred entries $(*)$ may have any value (including zero).

## 数学代写|线性代数代写linear algebra代考|Solutions of Linear Systems

The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system.
Suppose, for example, that the augmented matrix of a linear system has been changed into the equivalent reduced echelon form
$$\left[\begin{array}{rrrr} 1 & 0 & -5 & 1 \ 0 & 1 & 1 & 4 \ 0 & 0 & 0 & 0 \end{array}\right]$$
There are three variables because the augmented matrix has four columns. The associated system of equations is
$$\begin{array}{r} x_1-5 x_3=1 \ x_2+x_3=4 \ 0=0 \end{array}$$
The variables $x_1$ and $x_2$ corresponding to pivot columns in the matrix are called basic variables. ${ }^2$ The other variable, $x_3$, is called a free variable.

Whenever a system is consistent, as in (4), the solution set can be described explicitly by solving the reduced system of equations for the basic variables in terms of the free variables. This operation is possible because the reduced echelon form places each basic variable in one and only one equation. In (4), solve the first equation for $x_1$ and the second for $x_2$. (Ignore the third equation; it offers no restriction on the variables.)
$$\left{\begin{array}{l} x_1=1+5 x_3 \ x_2=4-x_3 \ x_3 \text { is free } \end{array}\right.$$
The statement ” $x_3$ is free” means that you are free to choose any value for $x_3$. Once that is done, the formulas in (5) determine the values for $x_1$ and $x_2$. For instance, when $x_3=0$, the solution is $(1,4,0)$; when $x_3=1$, the solution is $(6,3,1)$. Each different choice of $x_3$ determines a (different) solution of the system, and every solution of the system is determined by a choice of $x_3$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|ROW REDUCTION AND ECHELON FORMS

$1.1$ 节的“三角”矩阵，如

## 数学代写|线性代数代写linear algebra代考|Solutions of Linear Systems

$$\left[\begin{array}{llllllllllll} 1 & 0 & -5 & 1 & 0 & 1 & 1 & 4 & 0 & 0 & 0 & 0 \end{array}\right]$$

$$x_1-5 x_3=1 x_2+x_3=40=0$$

x_1=1+5 x_3 x_2=4-x_3 x_3 \text { is free }
$$正确的。 \ \$$

## 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|MATH1051

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。

• 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|线性代数代写linear algebra代考|Solving a Linear System

This section and the next describe an algorithm, or a systematic procedure, for solving linear systems. The basic strategy is to replace one system with an equivalent system (i.e., one with the same solution set) that is easier to solve.

Roughly speaking, use the $x_1$ term in the first equation of a system to eliminate the $x_1$ terms in the other equations. Then use the $x_2$ term in the second equation to eliminate the $x_2$ terms in the other equations, and so on, until you finally obtain a very simple equivalent system of equations.

Three basic operations are used to simplify a linear system: Replace one equation by the sum of itself and a multiple of another equation, interchange two equations, and multiply all the terms in an equation by a nonzero constant. After the first example, you will see why these three operations do not change the solution set of the system.

Row operations can be applied to any matrix, not merely to one that arises as the augmented matrix of a linear system. Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other.
It is important to note that row operations are reversible. If two rows are interchanged, they can be returned to their original positions by another interchange. If a row is scaled by a nonzero constant $c$, then multiplying the new row by $1 / c$ produces the original row. Finally, consider a replacement operation involving two rows -say, rows 1 and 2 -and suppose that $c$ times row 1 is added to row 2 to produce a new row 2. To “reverse” this operation, add $-c$ times row 1 to (new) row 2 and obtain the original row 2. See Exercises $29-32$ at the end of this section.

At the moment, we are interested in row operations on the augmented matrix of a system of linear equations. Suppose a system is changed to a new one via row operations. By considering each type of row operation, you can see that any solution of the original system remains a solution of the new system. Conversely, since the original system can be produced via row operations on the new system, each solution of the new system is also a solution of the original system. This discussion justifies the following statement.

## 数学代写|线性代数代写linear algebra代考|Existence and Uniqueness Questions

Section $1.2$ will show why a solution set for a linear system contains either no solutions, one solution, or infinitely many solutions. Answers to the following two questions will determine the nature of the solution set for a linear system.

To determine which possibility is true for a particular system, we ask two questions.

These two questions will appear throughout the text, in many different guises. This section and the next will show how to answer these questions via row operations on the augmented matrix.
EXAMPLE 2 Determine if the following system is consistent:
\begin{aligned} x_1-2 x_2+x_3 & =0 \ 2 x_2-8 x_3 & =8 \ 5 x_1-5 x_3 & =10 \end{aligned}
SOLUTION This is the system from Example 1. Suppose that we have performed the row operations necessary to obtain the triangular form
\begin{aligned} x_1-2 x_2+x_3 & =0 \ x_2-4 x_3 & =4 \ x_3 & =-1 \end{aligned} \quad\left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \ 0 & 1 & -4 & 4 \ 0 & 0 & 1 & -1 \end{array}\right] At this point, we know $x_3$. Were we to substitute the value of $x_3$ into equation 2 , we could compute $x_2$ and hence could determine $x_1$ from equation 1 . So a solution exists; the system is consistent. (In fact, $x_2$ is uniquely determined by equation 2 since $x_3$ has only one possible value, and $x_1$ is therefore uniquely determined by equation 1 . So the solution is unique.)

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Existence and Uniqueness Questions

$$x_1-2 x_2+x_3=02 x_2-8 x_3 \quad=85 x_1-5 x_3=10$$

$$x_1-2 x_2+x_3=0 x_2-4 x_3 \quad=4 x_3=-1 \quad\left[\begin{array}{llllllllllll} 1 & -2 & 1 & 0 & 0 & 1 & -4 & 4 & 0 & 0 & 1 & -1 \end{array}\right]$$

## 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|MATH1014

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。

• 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|线性代数代写linear algebra代考|SYSTEMS OF LINEAR EQUATIONS

A linear equation in the variables $x_1, \ldots, x_n$ is an equation that can be written in the form
$$a_1 x_1+a_2 x_2+\cdots+a_n x_n=b$$
where $b$ and the coefficients $a_1, \ldots, a_n$ are real or complex numbers, usually known in advance. The subscript $n$ may be any positive integer. In textbook examples and exercises, $n$ is normally between 2 and 5 . In real-life problems, $n$ might be 50 or 5000 , or even larger.
The equations
$$4 x_1-5 x_2+2=x_1 \quad \text { and } \quad x_2=2\left(\sqrt{6}-x_1\right)+x_3$$
are both linear because they can be rearranged algebraically as in equation (1):
$$3 x_1-5 x_2=-2 \text { and } 2 x_1+x_2-x_3=2 \sqrt{6}$$
The equations
$$4 x_1-5 x_2=x_1 x_2 \quad \text { and } \quad x_2=2 \sqrt{x_1}-6$$
are not linear because of the presence of $x_1 x_2$ in the first equation and $\sqrt{x_1}$ in the second. A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables-say, $x_1, \ldots, x_n$. An example is
$$\begin{array}{r} 2 x_1-x_2+1.5 x_3=8 \ x_1-4 x_3=-7 \end{array}$$ A solution of the system is a list $\left(s_1, s_2, \ldots, s_n\right)$ of numbers that makes each equation a true statement when the values $s_1, \ldots, s_n$ are substituted for $x_1, \ldots, x_n$, respectively. For instance, $(5,6.5,3)$ is a solution of system ( 2 ) because, when these values are substituted in (2) for $x_1, x_2, x_3$, respectively, the equations simplify to $8=8$ and $-7=-7$.

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

The essential information of a linear system can be recorded compactly in a rectangular array called a matrix. Given the system
\begin{aligned} x_1-2 x_2+x_3 & =0 \ 2 x_2-8 x_3 & =8 \ 5 x_1-5 x_3 & =10 \end{aligned}
with the coefficients of each variable aligned in columns, the matrix
$$\left[\begin{array}{rrr} 1 & -2 & 1 \ 0 & 2 & -8 \ 5 & 0 & -5 \end{array}\right]$$
is called the coefficient matrix (or matrix of coefficients) of the system (3), and
$$\left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \ 0 & 2 & -8 & 8 \ 5 & 0 & -5 & 10 \end{array}\right]$$
is called the augmented matrix of the system. (The second row here contains a zero because the second equation could be written as $0 \cdot x_1+2 x_2-8 x_3=8$.) An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations.

The size of a matrix tells how many rows and columns it has. The augmented matrix (4) above has 3 rows and 4 columns and is called a $3 \times 4$ (read “3 by 4 “) matrix. If $m$ and $n$ are positive integers, an $\boldsymbol{m} \times \boldsymbol{n}$ matrix is a rectangular array of numbers with $m$ rows and $n$ columns. (The number of rows always comes first.) Matrix notation will simplify the calculations in the examples that follow.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|SYSTEMS OF LINEAR EQUATIONS

$$a_1 x_1+a_2 x_2+\cdots+a_n x_n=b$$

$$4 x_1-5 x_2+2=x_1 \quad \text { and } \quad x_2=2\left(\sqrt{6}-x_1\right)+x_3$$

$$3 x_1-5 x_2=-2 \text { and } 2 x_1+x_2-x_3=2 \sqrt{6}$$

$$4 x_1-5 x_2=x_1 x_2 \quad \text { and } \quad x_2=2 \sqrt{x_1}-6$$

$$2 x_1-x_2+1.5 x_3=8 x_1-4 x_3=-7$$

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

$$x_1-2 x_2+x_3=02 x_2-8 x_3=85 x_1-5 x_3=10$$

$$\left[\begin{array}{llllllll} 1 & -2 & 1 & 0 & 2 & -85 & 0 & -5 \end{array}\right]$$

$$\left[\begin{array}{lllllllllll} 1 & -2 & 1 & 0 & 0 & 2 & -8 & 85 & 0 & -5 & 10 \end{array}\right]$$

## 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|MTH2106

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。

• 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|线性代数代写linear algebra代考|MATRIX MULTIPLICATION

Here, we present another operation applicable in $M_{m n}$ in which the inputs are two matrices and the output is another matrix. Although this is not an operation indicative of a vector space, it is an essential ingredient in what will follow.

Definition $1.11$ Let $A=\left[a_{i j}\right] \in M_{m n}$ and $B=\left[b_{i j}\right] \in M_{n r}$. Then the product $C=\left[c_{i j}\right]=A B \in M_{m r}$ is defined as follows:
$$c_{i j}=\sum_{k=1}^n a_{i k} b_{k j} .$$
Notice that to perform matrix multiplication on matrices, it is necessary that the number of columns in $A$ be equal to the number of rows in $B$ and the resulting matrix has the same number of rows as $A$ and the same number of columns as $B$. Perhaps a simpler way to remember the entries of $C$ is that the ijth entry of $C$ is obtained by taking the dot product of the $i$ th row of $A$ with the $j$ th column of $B$. Conversely, one can define dot product in terms of matrix multiplication. Indeed, if $v, w \in \mathbb{R}^n$, then $v \cdot w=v^T w$, where $v$ and $w$ are viewed as $n \times 1$ column matrices. This is sometimes a useful representation of dot product when demonstrating certain proofs.
Example $1.10$
$$\left[\begin{array}{lll} 1 & 2 & 3 \ 4 & 5 & 6 \end{array}\right]\left[\begin{array}{rrr} 1 & -1 & 1 \ -1 & 0 & 1 \ 0 & 1 & 1 \end{array}\right]$$
$$=\left[\begin{array}{lll} (1)(1)+(2)(-1)+(3)(0) & (1)(-1)+(2)(0)+(3)(1) & (1)(1)+(2)(1)+(3)(1) \ (4)(1)+(5)(-1)+(6)(0) & (4)(-1)+(5)(0)+(6)(1) & (4)(1)+(5)(1)+(6)(1) \end{array}\right]$$
$$=\left[\begin{array}{rrr} -1 & 2 & 6 \ -1 & 2 & 15 \end{array}\right]$$

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

We are ready to present a systematic way for solving systems of linear equations. This method is simple and will be used quite regularly throughout the remainder of the book. First, recall that every system of linear equations has an associated augmented matrix:
Example 2.2 The augmented matrix associated with the linear system
$$\left{\begin{array}{rlr} 2 x_1+x_2-x_3 & =0 \ x_1-3 x_2+x_3 & =7 \ -3 x_1+x_2+x_3 & = & -5 \end{array}\right.$$
is
$$\left[\begin{array}{rrr|r} 2 & 1 & -1 & 0 \ 1 & -3 & 1 & 7 \ -3 & 1 & 1 & -5 \end{array}\right]$$
In solving a linear system we wish to manipulate the equations without altering the solution set and arrive at a more “desirable” system of equations for which we can readily identify the solution set. The operations below achieve this goal.

Definition 2.3 The following three operations are called elementary row operations which can be applied to a system of linear equations or the associated augmented matrix:

1. Multiplying the ith equation (or ith row of the augmented matrix) by a non-zero scalar $a$. The notation is a$R_i$.
2. Switching the $i$ th and $j$ th equation (or ith and $j$ th row of the augmented matrix). The notation is $R_i \leftrightarrow R_j$.
3. Adding a scalar a times the ith equation to the $j$ th equation (or adding a times the ith row to the $j$ th row of the augmented matrix). The notation is a $R_i+R_j$.

# 线性代数代考

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

$$c_{i j}=\sum_{k=1}^n a_{i k} b_{k j} .$$

$v \cdot w=v^T w$ ，在哪里 $v$ 和 $w$ 被视为 $n \times 1$ 列矩阵。在演示某些证明时，这有时是点积的有用 表示。

\begin{aligned} & =[(1)(1)+(2)(-1)+(3)(0) \quad(1)(-1)+(2)(0)+(3)(1) \quad(1)(1)+(2)(1)+(3) \ & \end{aligned}

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

$\$ \$$Veft {$$
2 x_1+x_2-x_3=0 x_1-3 x_2+x_3=7-3 x_1+x_2+x_3=-5
$$、正确的。 is 剩下[$$
\begin{array}{lll|l|l|l|ll|l|l|l}
2 & 1 & -1 & 0 & 1 & -3 & 1 & 7-3 & 1 & 1 & -5
\end{array}
$$Iright] \ \$$

1. 将第 $\mathrm{i}$ 个方程 (或增广矩阵的第 $\mathrm{i}$ 行) 乘以非零标量 $a$. 该符号是 $R_i$.
2. 切换 $i$ 和 $j$ 第方程 (或第 $\mathrm{i}$ 和 $j$ 增广矩阵的第 th 行) 。符号是 $R_i \leftrightarrow R_j$.
3. 添加一个标量 $a$ 乘以第 $\mathrm{i}$ 个方程到 $j$ th 等式 (或将第 $\mathrm{i}$ 行的 $a$ 乘以 $j$ 增广矩阵的第 th 行)。该符号是 $R_i+R_j$.

## 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|MATH1051

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。

• 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|线性代数代写linear algebra代考|APPLICATION: GEOMETRY

As we have already stated tuples in $\mathbb{R}^n$ along with their operations take on a geometric meaning. This section is devoted to further exploration of this observation. Recall briefly the following geometric facts about tuples:

1. A vector, $u$, can be viewed physically as an arrow.
2. The sum and difference of two vectors, $u+v$ and $u-v$, comprise the diagonals of a parallelogram whose adjacent sides are these two vectors.
3. The magnitude of a vector, $|u|$, corresponds to the length of the arrow representing $u$.
4. For vectors $u$ and $v$, we have the equation $u \cdot v=|u||v| \cos \theta$, where $\theta$ is the smaller angle between $u$ and $v$.
5. Two vectors $u$ and $v$ are parallel iff $u=a v$ or $v=a u$ for some real number $a$.
6. Two vectors $u$ and $v$ are perpendicular iff $u \cdot v=0$.
7. The vector $-u$ points in the opposite direction of $u$.
Only in this section will we allow vectors which do not have their initial point at the origin so that we can derive some nice geometric results. In this case, we will say that two vectors are equal if they have the same length and are point in the same direction.

For instance, in Figure 1.5 we have depicted a collection of vectors which are all equal to each other.

We need to introduce some notation. If $A$ and $B$ are points in space, then $\overrightarrow{A B}$ denotes the vector with initial point $A$ and terminal point $B$ as depicted in Figure 1.6.
From our discussion of the parallelogram earlier, it is clear that if $u-$ $\left[a_1, a_2, \ldots, a_n\right]$ is a vector with terminal point at $A$ and $v=\left[b_1, b_2, \ldots, b_n\right]$ is a vector with terminal point at $B$, then
$$\overrightarrow{A B}=v-u=\left[b_1-a_1, b_2-a_2, \ldots, b_n-a_n\right] .$$
With just these few facts we are capable of proving many standard geometric results.

## 数学代写|线性代数代写linear algebra代考|SECOND VECTOR SPACE: MATRICES

Here now is our second example of what later will be called a vector space. First we define a matrix.

Definition $1.8$ An $m \times n$ matrix is a rectangular array of scalars consisting of $m$ rows and $n$ columns. We say the dimensions of the matrix are ” $m$-by- $n$ or $m \times n$. .”
Example $1.8\left[\begin{array}{rrr}-1 & \pi & 6 \ \sqrt{3} & -1.2 & 3 / 4\end{array}\right]$ is an example of a $2 \times 3$ matrix.
There are several useful ways of representing a matrix. The most descriptive (and most cumbersome) is the following:
$$\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \vdots & \vdots & \ddots & \vdots \ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right]$$
Each scalar $a_{i j}$ is called the $i j$ th entry of the matrix where $1 \leq i \leq m$ and $1 \leq j \leq n$. A simpler notation for a matrix is $\left[a_{i j}\right]$. We often represent a matrix simply by $A$. Another useful way to represent a matrix is by its rows or by its columns:
$$A=\left[\begin{array}{c} r_1 \ r_2 \ \vdots \ r_m \end{array}\right], \text { where } r_i=\left[\begin{array}{llll} a_{i 1} & a_{i 2} & \cdots & a_{i n} \end{array}\right] \quad(i=1,2, \ldots, m), \text { or }$$

$$A=\left[\begin{array}{llll} c_1 & c_2 & \cdots & c_n \end{array}\right] \text {, where } c_j=\left[\begin{array}{c} a_{1 j} \ a_{2 j} \ \vdots \ a_{m j} \end{array}\right] \quad(j=1,2, \ldots, n) .$$
We are now ready to define our second vector space.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|APPLICATION: GEOMETRY

1. 一个向量, $u$, 在物理上可以看作是一个箭头。
2. 两个向量的和与差， $u+v$ 和 $u-v$ ，包括平行四边形的对角线，其相邻边是这两个向 量。
3. 矢量的大小， $|u|$, 对应于代表箭头的长度 $u$.
4. 对于载体 $u$ 和 $v$ ，我们有方程 $u \cdot v=|u||v| \cos \theta$ ，在哪里 $\theta$ 是之间的较小角度 $u$ 和 $v$.
5. 两个向量 $u$ 和 $v$ 是平行的当且仅当 $u=a v$ 要么 $v=a u$ 对于一些实数 $a$.
6. 两个向量 $u$ 和 $v$ 是垂直的当且仅当 $u \cdot v=0$.
7. 载体 $-u$ 指向相反的方向 $u$.
仅在本节中，我们将允许初始点不在原点的向量，以便我们可以得出一些不错的几何结 果。在这种情况下，如果两个向量具有相同的长度并且指向相同的方向，我们就说它们 相等。
例如，在图 $1.5$ 中，我们描绘了一组彼此相等的向量。
我们需要引入一些符号。如果 $A$ 和 $B$ 是空间中的点，那么 $\overrightarrow{A B}$ 表示具有初始点的向量 $A$ 和终点 $B$ 如图 1.6 所示。
从我们之前对平行四边形的讨论中可以清楚地看出，如果 $u-\left[a_1, a_2, \ldots, a_n\right]$ 是一个向量， 其终点位于 $A$ 和 $v=\left[b_1, b_2, \ldots, b_n\right]$ 是一个向量，其终点位于 $B$ ，然后
$$\overrightarrow{A B}=v-u=\left[b_1-a_1, b_2-a_2, \ldots, b_n-a_n\right] .$$
仅凭这几个事实，我们就能够证明许多标准的几何结果。

## 数学代写|线性代数代写linear algebra代考|SECOND VECTOR SPACE: MATRICES

$A=\left[\begin{array}{llll}r_1 r_2 & \vdots & r_m\end{array}\right]$, where $r_i=\left[\begin{array}{llll}a_{i 1} & a_{i 2} & \cdots & a_{i n}\end{array}\right] \quad(i=1,2, \ldots, m)$, or
$$A=\left[\begin{array}{llll} c_1 & c_2 & \cdots & c_n \end{array}\right], \text { where } c_j=\left[\begin{array}{c} a_{1 j} a_{2 j} \vdots a_{m j} \end{array}\right] \quad(j=1,2, \ldots, n) .$$

## 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|MATH1014

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。

• 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|线性代数代写linear algebra代考|FIRST VECTOR SPACE: TUPLES

Here now is our first example of what later will be called a vector space. A notion in linear algebra of some importance is the scalar. For most of our discussion, a scalar will just be a real number and, at times, a complex number. A more comprehensive and perhaps advanced treatise on linear algebra would assume a scalar to be a element of what is called a field. Roughly speaking, a field gathers together some of the essential properties (or axioms) of the real numbers. We list these properties below:
Definition $1.1$ A field is a set of objects $F$ together with two operations $+$ and . (called addition and multiplication) having the following properties:
Closure: For all $a, b \in F$, we have $a+b \in F$ and $a \cdot b \in F$.
Commutativity: For all $a, b \in F$, we have $a+b=b+a$ and $a \cdot b=b \cdot a$.
Associativity: For all $a, b, c \in F$, we have $a+(b+c)=(a+b)+c$ and $a \cdot(b \cdot c)=(a \cdot b) \cdot c$.

Identity: There exist $0,1 \in F$ such that for all $a \in F$, we have $a+0-a$ and $a \cdot 1=a$.

Inverse: For every $a \in F$ there exists $b \in F$ such that $a+b=0$ ( $b$ is called the additive inverse of a) and for every $0 \neq a \in F$ there exists $b \in F$ such that $a \cdot b=1$ ( $b$ is called the multiplicative inverse of $a$ ).
Distribution: For all $a, b, c \in F$, we have $a \cdot(b+c)=a \cdot b+a \cdot c$.
The main examples of fields addressed in this text are the real numbers and the complex numbers (one can easily check that the properties above are satisfied in each example). At times we may want to prove results in more generality without assuming what field we have, but as stated, a scalar for the time being is simply another name for a real number. The standard notation for real numbers is $\mathbb{R}$.

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

Here we present another operation applicable in $\mathbb{R}^n$ in which the inputs are two vectors and the output is a scalar. The various names of this operation are dot, scalar or inner product. Although this is not an operation indicative of a vector space, it is an essential ingredient of what we will later call an inner product space.

Definition 1.4 Let $u=\left[a_1, \ldots, a_n\right], v=\left[b_1, \ldots, b_n\right] \in \mathbb{R}^n$. The dot product of $u$ and $v$, written
$$u \cdot v=a_1 b_1+\cdots+a_n b_n .$$
Example $1.3$ In $\mathbb{R}^4$,
$$\begin{gathered} {[2,25,-1,-1.3] \cdot[-3,1 / 5,3,10]=(2)(-3)+(25)(1 / 5)+(-1)(3)+(-1.3)(10)} \ =-6+5-3-13=-17 . \end{gathered}$$
The following result summarizes some elementary properties of the dot product:
Theorem 1.2 If $u, v, w \in \mathbb{R}^n$ and $a \in \mathbb{R}$, then
i. $u \cdot v=v \cdot u$.
ii. $u \cdot(v+w)=u \cdot v+u \cdot w$.
iii. $a(u \cdot v)=(a u) \cdot v=u \cdot(a v)$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|FIRST VECTOR SPACE: TUPLES

$a \cdot(b \cdot c)=(a \cdot b) \cdot c$.

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

$$u \cdot v=a_1 b_1+\cdots+a_n b_n .$$

$$[2,25,-1,-1.3] \cdot[-3,1 / 5,3,10]=(2)(-3)+(25)(1 / 5)+(-1)(3)+(-1.3)(10)$$

\begin{aligned} & \text { 二. } u \cdot(v+w)=u \cdot v+u \cdot w \ & \text { 三. } a(u \cdot v)=(a u) \cdot v=u \cdot(a v) \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|MTH2106

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。

• 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|线性代数代写linear algebra代考|The Geometry of Systems of Equations

It turns out that there is an intimate connection between solutions to systems of equations in two variables and the geometry of lines in $\mathbb{R}^2$. We recall the graphical method to solving systems below. Although you will likely have already done this in previous classes, we include it here so that you can put this knowledge into the context of solution sets to systems of equations as classified in Theorem 2.2.20.
We begin with the following simple example:
Example 2.2.27 Let us consider $u=\left(\begin{array}{c}2 \ -3\end{array}\right), v=\left(\begin{array}{l}1 \ 1\end{array}\right)$, and $w=\left(\begin{array}{l}2 \ 3\end{array}\right) \in \mathbb{R}^2$. Suppose we want to know if we can express $u$ using arithmetic operations on $v$ and $w$. In other words, we want to know if there are scalars $x, y$ so that
$$\left(\begin{array}{c} 2 \ -3 \end{array}\right)=x \cdot\left(\begin{array}{l} 1 \ 1 \end{array}\right)+y \cdot\left(\begin{array}{l} 2 \ 3 \end{array}\right) .$$
We can rewrite the right-hand side of the vector equation so that we have the equation with two vectors
$$\left(\begin{array}{c} 2 \ -3 \end{array}\right)=\left(\begin{array}{l} x+2 y \ x+3 y \end{array}\right) .$$
The equivalent system of linear equations with 2 equations and 2 variables is
\begin{aligned} & x+2 y=2 \ & x+3 y=-3 . \end{aligned}
Equations (2.18) and (2.19) are equations of lines in $\mathbb{R}^2$, that is, the set of pairs $(x, y)$ that satisfy each equation is the set of points on each respective line. Hence, finding $x$ and $y$ that satisfy both equations amounts to finding all points $(x, y)$ that are on both lines. If we graph these two lines, we can see that they appear to cross at one point, $(12,-5)$, and nowhere else, so we estimate $x=12$ and $y=-5$ is the only solution of the two equations. (See Figure 2.9.) You can also algebraically verify that $(12,5)$ is a solution to the system.

## 数学代写|线性代数代写linear algebra代考|Images and Image Arithmetic

In Section $2.1$ we saw that if you add two images, you get a new image, and that if you multiply an image by a scalar, you get a new image. We represented a rectangular pixelated image as an array of values, or equivalently, as a rectangular array of grayscale patches. This is a very natural idea in the context of digital photography.

Recall the definition of an image given in Section 2.1. We repeat it here, and follow the definition by some examples of images with different geometric arrangements.

An image is a finite ordered list of real values with an associated geometric arrangement.
Four examples of arrays along with an index system specifying the order of patches can be seen in Figure 2.11. As an image, each patch would also have a numerical value indicating the brightness of the patch (not shown in the figure). The first is a regular pixel array commonly used for digital photography. The second is a hexagonal pattern which also nicely tiles a plane. The third is a map of the African continent and Madagascar subdivided by country. The fourth is a square pixel set with enhanced resolution toward the center of the field of interest. It should be clear from the definition that images are not matrices. Only the first example might be confused with a matrix.

We first fix a particular geometric arrangement of pixels (and let $n$ denote the number of pixels in the arrangement). Then an image is precisely described by its (ordered) intensity values. With this determined, we formalize the notions of scalar multiplication and addition on images that were developed in the previous section.

Given two images $x$ and $y$ with (ordered) intensity values $\left(x_1, x_2, \cdots, x_n\right)$ and $\left(y_1, y_2, \cdots, y_n\right)$, respectively, and the same geometry, the image sum written $z=x+y$ is the image with intensity values $z_i=x_i+y_i$ for all $i \in{1,2, \cdots, n}$, and the same geometry.

Hence, the sum of two images is the image that results from pixel-wise addition of intensity values. Put another way, the sum of two images is the image that results from adding corresponding values of their ordered lists, while maintaining the geometric arrangement of pixels.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|The Geometry of Systems of Equations

$$\left(\begin{array}{c}) 2 \ -3 \end{array}right）=x\cdot\left(begin{array}{l}) 1 \ 1 \end{array}right）+y cdot\left(begin{array}{l}) 2 \ 3 \end{array}right）。$$

$$\left(\begin{array}{c}) 2 \ -3 \end{array}right）=left(begin{array}{l}) x+2 y x+3 y \end{array}right）。$$

\begin{aligned} & x+2 y=2 & x+3 y=-3 。 & x+3 y=-3 。 \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。