## 数学代写|代数学代写Algebra代考|MAT523

statistics-lab™ 为您的留学生涯保驾护航 在代写代数学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。我们的专家在代写代数学Algebra代写方面经验极为丰富，各种代写代数学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|代数学代写Algebra代考|The Dot Product

The main tool that helps us extend geometric notions from $\mathbb{R}^2$ and $\mathbb{R}^3$ to arbitrary dimensions is the dot product, which is a way of combining two vectors so as to create a single number:

Suppose $\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right) \in \mathbb{R}^n$ and $\mathbf{w}=\left(w_1, w_2, \ldots, w_n\right) \in \mathbb{R}^n$ are vectors. Then their dot product, denoted by $\mathbf{v} \cdot \mathbf{w}$, is the quantity
$$\mathbf{v} \cdot \mathbf{w} \stackrel{\text { dff }}{=} v_1 w_1+v_2 w_2+\cdots+v_n w_n .$$
It is important to keep in mind that the output of the dot product is a number, not a vector. So, for example, the expression $\mathbf{v} \cdot(\mathbf{w} \cdot \mathbf{x})$ does not make sense, since $\mathbf{w} \cdot \mathbf{x}$ is a number, and so we cannot take its dot product with $\mathbf{v}$. On the other hand, the expression $\mathbf{v} /(\mathbf{w} \cdot \mathbf{x})$ does make sense, since dividing a vector by a number is a valid mathematical operation. As we introduce more operations between different types of objects, it will become increasingly important to keep in mind the type of object that we are working with at all times.

Compute (or state why it’s impossible to compute) the following dot products:
a) $(1,2,3) \cdot(4,-3,2)$,
b) $(3,6,2) \cdot(-1,5,2,1)$, and c) $\left(v_1, v_2, \ldots, v_n\right) \cdot \mathbf{e}j$, where $1 \leq j \leq n$. Solutions: a) $(1,2,3) \cdot(4,-3,2)=1 \cdot 4+2 \cdot(-3)+3 \cdot 2=4-6+6=4$. b) $(3,6,2) \cdot(-1,5,2,1)$ does not exist, since these vectors do not have the same number of entries. c) For this dot product to make sense, we have to assume that the vector $\mathbf{e}_j$ has $n$ entries (the same number of entries as $\left(v_1, v_2, \ldots, v_n\right)$ ). Then \begin{aligned} \left(v_1, v_2, \ldots, v_n\right) \cdot \mathbf{e}_j &=0 v_1+\cdots+0 v{j-1}+1 v_j+0 v_{j+1}+\cdots+0 v_n \ &=v_j . \end{aligned}
The dot product can be interpreted geometrically as roughly measuring the amount of overlap between $\mathbf{v}$ and $\mathbf{w}$. For example, if $\mathbf{v}=\mathbf{w}=(1,0)$ then $\mathbf{v} \cdot \mathbf{w}=1$, but as we rotate $\mathbf{w}$ away from $\mathbf{v}$, their dot product decreases down to 0 when $\mathbf{v}$ and $\mathbf{w}$ are perpendicular (i.e., when $\mathbf{w}=(0,1)$ or $\mathbf{w}=(0,-1))$, as illustrated in Figure 1.7. It then decreases even farther down to $-1$ when $w$ points in the opposite direction of $\mathbf{v}$ (i.e., when $\mathbf{w}=(-1,0)$ ).

More specifically, if we rotate $w$ counter-clockwise from $\mathbf{v}$ by an angle of $\theta$ then its coordinates become $w=(\cos (\theta), \sin (\theta))$. The dot product between $\mathbf{v}$ and $\mathbf{w}$ is then $\mathbf{v} \cdot \mathbf{w}=1 \cos (\theta)+0 \sin (\theta)=\cos (\theta)$, which is largest when $\theta$ is small (i.e., when w points in almost the same direction as $\mathbf{v}$ ).

## 数学代写|代数学代写Algebra代考|The Angle Between Vectors

In order to get a bit of an idea of how to discuss the angle between vectors in terms of things like the dot product, we first focus on vectors in $\mathbb{R}^2$ or $\mathbb{R}^3$. In these lower-dimensional cases, we can use geometric techniques to determine the angle between two vectors $\mathbf{v}$ and $\mathbf{w}$. If $\mathbf{v}, \mathbf{w} \in \mathbb{R}^2$ then we can place $\mathbf{v}$ and $\mathbf{w}$ in standard position, so that the vectors $\mathbf{v}, \mathbf{w}$, and $\mathbf{v}-\mathbf{w}$ form the sides of a triangle, as in Figure 1.11(a).

We can then use the law of cosines to relate $|\mathbf{v}|,|\mathbf{w}|,|\mathbf{v}-\mathbf{w}|$, and the angle $\theta$ between $\mathbf{v}$ and $\mathbf{w}$. Specifically, we find that
$$|\mathbf{v}-\mathbf{w}|^2=|\mathbf{v}|^2+|\mathbf{w}|^2-2|\mathbf{v}||\mathbf{w}| \cos (\theta) .$$
On the other hand, the basic properties of the dot product that we saw back in Theorem 1.2.1 tell us that
\begin{aligned} |\mathbf{v}-\mathbf{w}|^2 &=(\mathbf{v}-\mathbf{w}) \cdot(\mathbf{v}-\mathbf{w}) \ &=\mathbf{v} \cdot \mathbf{v}-\mathbf{v} \cdot \mathbf{w}-\mathbf{w} \cdot \mathbf{v}+\mathbf{w} \cdot \mathbf{w}=|\mathbf{v}|^2-2(\mathbf{v} \cdot \mathbf{w})+|\mathbf{w}|^2 \end{aligned}

By setting these two expressions for $|\mathbf{v}-\mathbf{w}|^2$ equal to each other, we see that
$$|\mathbf{v}|^2+|\mathbf{w}|^2-2|\mathbf{v}||\mathbf{w}| \cos (\theta)=|\mathbf{v}|^2-2(\mathbf{v} \cdot \mathbf{w})+|\mathbf{w}|^2 .$$
Simplifying and rearranging this equation then gives a formula for $\theta$ in terms of the lengths of $\mathbf{v}$ and $\mathbf{w}$ and their dot product:
$$\cos (\theta)=\frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}||\mathbf{w}|}, \quad \text { so } \quad \theta=\arccos \left(\frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}||\mathbf{w}|}\right) .$$
This argument still works, but is slightly trickier to visualize, when working with vector $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$ that are 3-dimensional. In this case, we can still arrange $\mathbf{v}, \mathbf{w}$, and $\mathbf{v}-\mathbf{w}$ to form a triangle, and the calculation that we did in $\mathbb{R}^2$ is the exact same – the only change is that the triangle is embedded in 3-dimensional space, as in Figure 1.11(b).

When considering vectors in higher-dimensional spaces, we no longer have a visual guide for what the angle between two vectors means, so instead we simply define the angle so as to be consistent with the formula that we derived above:
The angle $\theta$ between two non-zero vectors $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ is the quantity
$$\theta=\arccos \left(\frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}||\mathbf{w}|}\right)$$

## 数学代写|代数学代写Algebra代考|The Dot Product

$$\mathbf{v} \cdot \mathbf{w} \stackrel{\text { dff }}{=} v_1 w_1+v_2 w_2+\cdots+v_n w_n .$$
。重要的是要记住，点积的输出是一个数字，而不是一个向量。因此，例如，表达式$\mathbf{v} \cdot(\mathbf{w} \cdot \mathbf{x})$没有意义，因为$\mathbf{w} \cdot \mathbf{x}$是一个数字，所以我们不能取它与$\mathbf{v}$的点积。另一方面，表达式$\mathbf{v} /(\mathbf{w} \cdot \mathbf{x})$是有意义的，因为用一个数字除以一个向量是一个有效的数学运算。当我们在不同类型的对象之间引入更多的操作时，时刻记住我们正在处理的对象的类型将变得越来越重要

a) $(1,2,3) \cdot(4,-3,2)$，
b) $(3,6,2) \cdot(-1,5,2,1)$，和c) $\left(v_1, v_2, \ldots, v_n\right) \cdot \mathbf{e}j$，其中$1 \leq j \leq n$。a) $(1,2,3) \cdot(4,-3,2)=1 \cdot 4+2 \cdot(-3)+3 \cdot 2=4-6+6=4$。B) $(3,6,2) \cdot(-1,5,2,1)$不存在，因为这些向量没有相同数量的条目。c)为了使这个点积有意义，我们必须假设向量$\mathbf{e}_j$有$n$个条目(与$\left(v_1, v_2, \ldots, v_n\right)$的条目数量相同)。那么\begin{aligned} \left(v_1, v_2, \ldots, v_n\right) \cdot \mathbf{e}_j &=0 v_1+\cdots+0 v{j-1}+1 v_j+0 v_{j+1}+\cdots+0 v_n \ &=v_j . \end{aligned}

$$\theta=\arccos \left(\frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}||\mathbf{w}|}\right)$$

## 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|代数学代写Algebra代考|Math4120

statistics-lab™ 为您的留学生涯保驾护航 在代写代数学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。我们的专家在代写代数学Algebra代写方面经验极为丰富，各种代写代数学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|代数学代写Algebra代考|Scalar Multiplication

The other basic operation on vectors that we introduce at this point is one that changes a vector’s length and/or reverses its direction, but does not otherwise change the direction in which it points.

Suppose $\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right) \in \mathbb{R}^n$ is a vector and $c \in \mathbb{R}$ is a scalar. Then their scalar multiplication, denoted by $c \mathbf{v}$, is the vector
$$c \mathbf{v} \stackrel{\text { dff }}{=}\left(c v_1, c v_2, \ldots, c v_n\right) .$$
We remark that, once again, algebraically this is exactly the definition that someone would likely expect the quantity $c \mathbf{v}$ to have. Multiplying each entry of $\mathbf{v}$ by $c$ seems like a rather natural operation, and it has the simple geometric interpretation of stretching $\mathbf{v}$ by a factor of $c$, as in Figure 1.4. In particular, if $|c|>1$ then scalar multiplication stretches $\mathbf{v}$, but if $|c|<1$ then it shrinks $\mathbf{v}$. When $c<0$ then this operation also reverses the direction of $\mathbf{v}$, in addition to any stretching or shrinking that it does if $|c| \neq 1$.

Two special cases of scalar multiplication are worth pointing out:

• If $c=0$ then $c v$ is the zero vector, all of whose entries are 0 , which we denote by 0 .
• If $c=-1$ then $c \mathbf{v}$ is the vector whose entries are the negatives of $\mathbf{v}$ ‘s entries, which we denote by $-\mathbf{v}$.
We also define vector subtraction via $\mathbf{v}-\mathbf{w} \stackrel{\text { dif }}{=} \mathbf{v}+(-\mathbf{w})$, and we note that it has the geometric interpretation that $\mathbf{v}-\mathbf{w}$ is the vector pointing from the head of $\mathbf{w}$ to the head of $\mathbf{v}$ when $\mathbf{v}$ and $\mathbf{w}$ are in standard position. It is perhaps easiest to keep this geometric picture straight (“it points from the head of which vector to the head of the other one?”) if we just think of $\mathbf{v}-\mathbf{w}$ as the vector that must be added to $\mathbf{w}$ to get $\mathbf{v}$ (so it points from $\mathbf{w}$ to $\mathbf{v}$ ). Alternatively, $\mathbf{v}-\mathbf{w}$ is the other diagonal (besides $\mathbf{v}+\mathbf{w}$ ) in the parallelogram with sides $\mathbf{v}$ and $\mathbf{w}$, as in Figure 1.5.
• It is straightforward to verify some simple properties of the zero vector, such as the facts that $\mathbf{v}-\mathbf{v}=\mathbf{0}$ and $\mathbf{v}+\mathbf{0}=\mathbf{v}$ for every vector $\mathbf{v} \in \mathbb{R}^n$, by working entry-by-entry with the vector operations. There are also quite a few other simple ways in which scalar multiplication interacts with vector addition, some of which we now list explicitly for easy reference.

## 数学代写|代数学代写Algebra代考|Linear Combinations

One common task in linear algebra is to start out with some given collection of vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k$ and then use vector addition and scalar multiplication to construct new vectors out of them. The following definition gives a name to this concept.

For example, $(1,2,3)$ is a linear combination of the vectors $(1,1,1)$ and $(-1,0,1)$ since $(1,2,3)=2(1,1,1)+(-1,0,1)$. On the other hand, $(1,2,3)$ is not a linear combination of the vectors $(1,1,0)$ and $(2,1,0)$ since every vector of the form $c_1(1,1,0)+c_2(2,1,0)$ has a 0 in its third entry, and thus cannot possibly equal $(1,2,3)$.

When working with linear combinations, some particularly important vectors are those with all entries equal to 0 , except for a single entry that equals 1 . Specifically, for each $j=1,2, \ldots, n$, we define the vector $\mathbf{e}_j \in \mathbb{R}^n$ by
$$\mathbf{e}_j \stackrel{\text { df }}{=}(0,0, \ldots, 0,1,0, \ldots, 0) .$$
For example, in $\mathbb{R}^2$ there are two such vectors: $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_2=(0,1)$. Similarly, in $\mathbb{R}^3$ there are three such vectors: $\mathbf{e}_1=(1,0,0), \mathbf{e}_2=(0,1,0)$, and $\mathbf{e}_3=(0,0,1)$. In general, in $\mathbb{R}^n$ there are $n$ of these vectors, $\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n$, and we call them the standard basis vectors (for reasons that we discuss in the next chapter). Notice that in $\mathbb{R}^2$ and $\mathbb{R}^3$, these are the vectors that point a distance of 1 in the direction of the $x-, y$-, and $z$-axes, as in Figure 1.6.

For now, the reason for our interest in these standard basis vectors is that every vector $\mathbf{v} \in \mathbb{R}^n$ can be written as a linear combination of them. In particular, if $\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right)$ then
$$\mathbf{v}=v_1 \mathbf{e}_1+v_2 \mathbf{e}_2+\cdots+v_n \mathbf{e}_n,$$
which can be verified just by computing each of the entries of the linear combination on the right. This idea of writing vectors in terms of the standard basis vectors (or other distinguished sets of vectors that we introduce later) is one of the most useful techniques that we make use of in linear algebra: in many situations, if we can prove that some property holds for the standard basis vectors, then we can use linear combinations to show that it must hold for all vectors.

## 数学代写|代数学代写Algebra代考|标量乘法

$$c \mathbf{v} \stackrel{\text { dff }}{=}\left(c v_1, c v_2, \ldots, c v_n\right) .$$我们注意到，再一次，从代数上讲，这正是某人可能期望的量的定义 $c \mathbf{v}$ 拥有。乘以的每一项 $\mathbf{v}$ 通过 $c$ 看起来是一个很自然的操作，它对拉伸有简单的几何解释 $\mathbf{v}$ 乘以 $c$，如图1.4所示。特别是，如果 $|c|>1$ 那么标量乘法就会延伸 $\mathbf{v}$，但如果 $|c|<1$ 然后收缩 $\mathbf{v}$。什么时候 $c<0$ 那么这个操作的方向也就颠倒了 $\mathbf{v}$除了它所做的任何拉伸或收缩 $|c| \neq 1$.

• $c=0$ 然后 $c v$ 是零向量，它的所有元素都是0，我们用0表示。
• If $c=-1$ 然后 $c \mathbf{v}$ 这个向量的分量是负数吗 $\mathbf{v}$ 的条目，我们用 $-\mathbf{v}$.
我们还通过定义向量减法 $\mathbf{v}-\mathbf{w} \stackrel{\text { dif }}{=} \mathbf{v}+(-\mathbf{w})$，我们注意到它的几何解释是 $\mathbf{v}-\mathbf{w}$ 向量是否指向的头部 $\mathbf{w}$ 到 $\mathbf{v}$ 何时 $\mathbf{v}$ 和 $\mathbf{w}$ 处于标准位置。也许最容易保持这个几何图形的直线(“它从哪个向量的头部指向另一个向量的头部?”)，如果我们只是想 $\mathbf{v}-\mathbf{w}$ 作为必须加到的向量 $\mathbf{w}$ 得到 $\mathbf{v}$ (所以它指向 $\mathbf{w}$ 到 $\mathbf{v}$ )。或者， $\mathbf{v}-\mathbf{w}$ 另一条对角线(除了? $\mathbf{v}+\mathbf{w}$ )在有边的平行四边形中 $\mathbf{v}$ 和 $\mathbf{w}$，如图1.5所示。
• 验证零向量的一些简单性质是很直接的，比如 $\mathbf{v}-\mathbf{v}=\mathbf{0}$ 和 $\mathbf{v}+\mathbf{0}=\mathbf{v}$ 对于每一个向量 $\mathbf{v} \in \mathbb{R}^n$，通过用向量运算进行逐入口运算。还有许多其他简单的方法可以使标量乘法与向量加法相互作用，我们现在显式列出其中一些方法，以方便参考
数学代写|代数学代写Algebra代考|线性组合
线性代数中的一个常见任务是，从某个给定的向量集合$\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k$开始，然后使用向量加法和标量乘法从它们中构造出新的向量。下面的定义给出了这个概念的名称例如，$(1,2,3)$是由$(1,2,3)=2(1,1,1)+(-1,0,1)$开始的向量$(1,1,1)$和$(-1,0,1)$的线性组合。另一方面，$(1,2,3)$不是向量$(1,1,0)$和$(2,1,0)$的线性组合，因为$c_1(1,1,0)+c_2(2,1,0)$形式的每个向量在第三个条目中都有一个0，因此不可能等于$(1,2,3)$当处理线性组合时，一些特别重要的向量是那些所有项都等于0的向量，只有一个项等于1。具体来说，对于每个$j=1,2, \ldots, n$，我们通过
$$\mathbf{e}_j \stackrel{\text { df }}{=}(0,0, \ldots, 0,1,0, \ldots, 0) .$$
来定义向量$\mathbf{e}_j \in \mathbb{R}^n$。例如，在$\mathbb{R}^2$中有两个这样的向量:$\mathbf{e}_1=(1,0)$和$\mathbf{e}_2=(0,1)$。类似地，在$\mathbb{R}^3$中有三个这样的向量:$\mathbf{e}_1=(1,0,0), \mathbf{e}_2=(0,1,0)$和$\mathbf{e}_3=(0,0,1)$。一般来说，在$\mathbb{R}^n$中有$n$这些向量，$\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n$，我们称它们为标准基向量(原因我们将在下一章讨论)。注意，在$\mathbb{R}^2$和$\mathbb{R}^3$中，这些是指向$x-, y$ -和$z$ -轴方向上距离为1的向量，如图1.6所示现在，我们对这些标准基向量感兴趣的原因是，每个向量$\mathbf{v} \in \mathbb{R}^n$都可以写成它们的线性组合。特别是，如果$\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right)$那么
$$\mathbf{v}=v_1 \mathbf{e}_1+v_2 \mathbf{e}_2+\cdots+v_n \mathbf{e}_n,$$
这可以通过计算右边线性组合的每一项来验证。这种用标准基向量(或我们稍后介绍的其他不同的向量集)来表示向量的想法是我们在线性代数中使用的最有用的技巧之一:在许多情况下，如果我们能证明某些性质适用于标准基向量，那么我们就可以使用线性组合来证明它一定适用于所有向量

## 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|代数学代写Algebra代考|MATH355

statistics-lab™ 为您的留学生涯保驾护航 在代写代数学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。我们的专家在代写代数学Algebra代写方面经验极为丰富，各种代写代数学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|代数学代写Algebra代考|Vectors and Vector Operations

In earlier math courses, focus was on how to manipulate expressions involving a single variable. For example, we learned how to solve equations like $4 x-3=7$ and we learned about properties of functions like $f(x)=3 x+8$, where in each case the one variable was called ” $x$ “. One way of looking at linear algebra is the natural extension of these ideas to the situation where we have two or more variables. For example, we might try solving an equation like $3 x+2 y=1$, or we might want to investigate the properties of a function that takes in two independent variables and outputs two dependent variables.

To make expressions involving several variables easier to deal with, we use vectors, which are ordered lists of numbers or variables. We say that the number of entries in the vector is its dimension, and if a vector has $n$ entries, we say that it “lives in” or “is an element of” $\mathbb{R}^n$. We denote vectors themselves by lowercase bold letters like $\mathbf{v}$ and $\mathbf{w}$, and we write their entries within parentheses. For example, $\mathbf{v}=(2,3) \in \mathbb{R}^2$ is a 2 -dimensional vector and $\mathbf{w}=(1,3,2) \in \mathbb{R}^3$ is a 3-dimensional vector (just like $4 \in \mathbb{R}$ is a real number).
In the 2 – and 3-dimensional cases, we can visualize vectors as arrows that indicate displacement in different directions by the amount specified in their entries. The vector’s first entry represents displacement in the $x$-direction, its second entry represents displacement in the $y$-direction, and in the 3-dimensional case its third entry represents displacement in the $z$-direction, as in Figure 1.1.
The front of a vector, where the tip of the arrow is located, is called its head, and the opposite end is called its tail. One way to compute the entries of a vector is to subtract the coordinates of its tail from the corresponding coordinates of its head. For example, the vector that goes from the point $(-1,1)$ to the point $(2,2)$ is $(2,2)-(-1,1)=(3,1)$. However, this is also the same as the vector that points from $(1,0)$ to $(4,1)$, since $(4,1)-(1,0)=(3,1)$ as well.

It is thus important to keep in mind that the coordinates of a vector specify its length and direction, but not its location in space; we can move vectors around in space without actually changing the vector itself, as in Figure 1.2. To remove this ambiguity when discussing vectors, we often choose to display them with their tail located at the origin – this is called the standard position of the vector.

Even though we can represent vectors in 2 and 3 dimensions via arrows, we emphasize that one of our goals is to keep vectors (and all of our linear algebra tools) as dimension-independent as possible. Our visualizations involving arrows can thus help us build intuition for how vectors behave, but our definitions and theorems themselves should work just as well in $\mathbb{R}^7$ (even though we cannot really visualize this space) as they do in $\mathbb{R}^3$. For this reason, we typically introduce new concepts by first giving the algebraic, dimension-independent definition, followed by some examples to illustrate the geometric significance of the new concept. We start with vector addition, the simplest vector operation that there is.

Vector addition can be motivated in at least two different ways. On the one hand, it is algebraically the simplest operation that could reasonably be considered a way of adding up two vectors: most students, if asked to add up two vectors, would add them up entry-by-entry even if they had not seen Definition 1.1.1. On the other hand, vector addition also has a simple geometric picture in terms of arrows: If $\mathbf{v}$ and $\mathbf{w}$ are positioned so that the tail of $\mathbf{w}$ is located at the same point as the head of $\mathbf{v}$ (in which case we say that $\mathbf{v}$ and $\mathbf{w}$ are positioned head-to-tail), then $\mathbf{v}+\mathbf{w}$ is the vector pointing from the tail of $\mathbf{v}$ to the head of $\mathbf{w}$, as in Figure 1.3(a). In other words, $\mathbf{v}+\mathbf{w}$ represents the total displacement accrued by following $\mathbf{v}$ and then following $\mathbf{w}$.

If we instead work entirely with vectors in standard position, then $\mathbf{v}+$ $\mathbf{w}$ is the vector that points along the diagonal between sides $\mathbf{v}$ and $\mathbf{w}$ of a parallelogram, as in Figure 1.3(b).

## 数学代写|代数学代写代数代考|向量与向量运算

. . 数学代写|代数学代写代数代考|

## 数学代写|代数学代写Algebra代考|Vector加法

.

$$R^{-1}=(y, x):(x, y) \in R \subseteq Y \times X .$$

(一) $(T \circ S) \circ R=T \circ(S \circ R)$ (关联财产) ；
(二) $(S \circ R)^{-1}=R^{-1} \circ S^{-1}$.

(ii) 再次 $S \circ R: X \rightarrow Z \Leftrightarrow(S \circ R)^{-1}: Z \rightarrow X$.

$(y, z) \in S \Leftrightarrow \exists y \in Y$ 这样 $(y, x) \in R^{-1}$ 和 $(z, y) \in S^{-1} \Leftrightarrow \exists y \in Y$ 这样 $(z, y) \in S^{-1}$ 和
$(y, x) \in R^{-1} \Leftrightarrow(z, x) \in R^{-1} \circ S^{-1}$.

## 数学代写|现代代数代写Modern Algebra代考|Functions or Mappings

$$(x, y) \in f \quad \text { and }(x, z) \in f \quad \text { imply } \quad y=z$$
(IE， $f$ 是单值)。

$\operatorname{dom} f=x \in X:(x, y) \in f$ for some $y \in Y \subseteq X ;$ range $f \quad=y \in Y:(x, y) \in f$ for some $x \in X$

## 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。

## 英国补考|现代代数代写Modern Algebra代考|Math 417

statistics-lab™ 为您的留学生涯保驾护航 在代写现代代数Modern 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。我们的专家在代写现代代数Modern Algebra代写方面经验极为丰富，各种代写现代代数Modern 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

## 英国补考|现代代数代写Modern Algebra代考|Equivalence Relation

A fundamental mathematical construction is to start with a non-empty set $X$ and to decompose the set into a family of disjoint subsets of $X$ whose union is the whole set $X$, called a partition of $X$ and to form a new set by equating each such subset to an element of a new set, called a quotient set of $X$ given by the partition. For this purpose we introduce the concept of an equivalence relation which is logically equivalent to a partition.

Definition 1.2.2 A binary relation $R$ on $A$ is said to be an equivalence relation on $A$ iff
(a) $R$ is reflexive: $(a, a) \in R$ for all $a \in A$;
(b) $R$ is symmetric: if $(a, b) \in R$, then $(b, a) \in R$ for $a, b \in A$;
(c) $R$ is transitive: if $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$ for $a, b, c \in A$.

Instead of speaking about subsets of $A \times A$, we can also define an equivalence relation as below by writing $a R b$ in place of $(a, b) \in R$.

Definition 1.2.3 A binary relation $R$ on $A$ is said to be an equivalence relation iff $R$ is
(a’) reflexive: $a R a$ for all $a \in A$;
(b’) symmetric: $a R b$ implies $b R a$ for $a, b \in A$;
(c’) transitive: $a R b$ and $b R c$ imply $a R c$ for $a, b, c \in A$.
Example 1.2.3 Define $R$ on $\mathbf{Z}$ by $a R b \Leftrightarrow a-b$ is divisible by a fixed integer $n>1$. Then $R$ is an equivalence relation.

Proof Since $a-a=0$ is divisible by $n$ for all $a \in \mathbf{Z}, a$ Ra for all $a \in \mathbf{Z}$, hence $R$ is reflexive. If $a-b$ is divisible by $n$, then $b-a$ is also divisible by $n$; hence $R$ is symmetric. Finally, if $a-b$ and $b-c$ are both divisible by $n$, then their sum $a-c$ is also divisible by $n$; hence $R$ is transitive. Consequently, $R$ is an equivalence relation. (See Example 1.2.9.)

## 英国补考|现代代数代写Modern Algebra代考|Partial Order Relations

We have made so far little use of reflexive, antisymmetric, and transitive laws. We are familiar with the natural ordering $\leq$ between two positive integers. This example suggests the abstract concept of a partial order relation, which is a reflexive, antisymmetric, and transitive relation. Partial order relations and their special types play an important role in mathematics. For example, partial order relations are essential in Zorn’s Lemma, which provides a very powerful tool in mathematics and in lattice theory whose applications are enormous in different sciences.

Definition 1.2.6 A reflexive, antisymmetric, and transitive relation $R$ on a nonempty set $P$ is called a partial order relation. Then the pair $(P, R)$ is called a partially ordered set or a poset.

We adopt the symbol ‘ $\leq$ ‘ to represent a partial order relation. So writing $a \leq b$ in place of $a R b$, from Definition $1.2 .6$ it follows that

(i) $a \leq a$ for all $a \in P$;
(ii) $a \leq b$ and $b \leq a$ in $P \Rightarrow a=b$ for $a, b \in P$ and
(iii) $a \leq b$ and $b \leq c$ in $P \Rightarrow a \leq c$ for $a, b, c \in P$.
The following three examples are quite different in nature but possess identical important properties.

## 英国补考|现代代数代写Modern Algebra代考|Equivalence Relation

$($ ( ) $R$ 是反身的: $(a, a) \in R$ 对所有人 $a \in A$;
(二) $R$ 是对称的: 如果 $(a, b) \in R$ ，然后 $(b, a) \in R$ 为了 $a, b \in A$;
(C) $R$ 是传递的: 如果 $(a, b) \in R$ 和 $(b, c) \in R$ ，然后 $(a, c) \in R$ 为了 $a, b, c \in A$.

(a’) 反身的: $a R a$ 对所有人 $a \in A$;
(b’) 对称: $a R b$ 暗示 $b R a$ 为了 $a, b \in A$ ；
(c’) 及物: $a R b$ 和 $b R c$ 意味着 $a R c$ 为了 $a, b, c \in A$.

## 英国补考|现代代数代写Modern Algebra代考|Partial Order Relations

(一世) $a \leq a$ 对所有人 $a \in P$;
(二) $a \leq b$ 和 $b \leq a$ 在 $P \Rightarrow a=b$ 为了 $a, b \in P$ (
iii) $a \leq b$ 和 $b \leq c$ 在 $P \Rightarrow a \leq c$ 为了 $a, b, c \in P$.

## 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|现代代数代写Modern Algebra代考|MATH 355

statistics-lab™ 为您的留学生涯保驾护航 在代写现代代数Modern 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。我们的专家在代写现代代数Modern Algebra代写方面经验极为丰富，各种代写现代代数Modern 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|现代代数代写Modern Algebra代考|Introductory Concepts

The concept of ‘set’ is very important in all branches of mathematics. We come across certain terms or concepts whose meanings need no explanation. Such terms are called undefined terms and are considered as primitive concepts. If one defines the term ‘set’ as ‘a set is a well defined collection of objects’, then the meaning of collection is not clear. One may define ‘a collection’ as ‘an aggregate’ of objects. What is the meaning of ‘aggregate’? As our language is finite, other synonyms, such as ‘class’, ‘family’ etc., will exhaust. Mathematicians accept that there are undefined terms and ‘set’ shall be such an undefined term. But we accept the familiar expressions, such as ‘set of all integers’, ‘set of all natural numbers’, ‘set of all rational numbers’, ‘set of all real numbers’ etc.

We shall neither attempt to give a formal definition of a set nor try to lay the groundwork for an axiomatic theory of sets. Instead we shall take the operational and intuitive approach to define a set. A set is a well defined collection of distinguishable objects.

The term ‘well defined’ specifies that it can be determined whether or not certain objects belong to the set in question. In most of our applications we deal with rather specific objects, and the nebulous notion of a set, in these, emerge as something quite recognizable. We usually denote sets by capital letters, such as $A, B, C, \ldots$. The objects of a set are called the elements or members of the set and are usually denoted by small letters, such as $a, b, c, \ldots$. Given a set $A$ we use the notation throughout ‘ $a \in A^{\prime}$ to indicate that an element $a$ is a member of $A$ and this is read as ‘ $a$ is an element of $A^{\prime}$ or ‘ $a$ belongs to $A$ ‘; and ‘ $a \notin A$ ‘ to indicate that the element $a$ is not a member of $A$ and this is read as ‘ $a$ is not an element of $A$ ‘ or ‘ $a$ does not belong to $A^{\prime}$. Since a set is uniquely determined by its elements, we may describe a set either by a characterizing property of the elements or by listing the elements. The standard way to describe a set by listing elements is to list elements of the set separated by commas, in braces. Thus a set $A={a, b, c}$ indicates that $a$, $b, c$ are the only elements of $A$ and nothing else. If $B$ is a set which consists of $a$, $b, c$ and possibly more, then notationally, $B={a, b, c, \ldots}$. On the other hand, a set consisting of a single element $x$ is sometimes called singleton $x$, denoted by ${x}$. By a statement, we mean a sentence about specific objects such that it has a truth value of either true or false but not both. If a set $A$ is described by a characterizing property $P(x)$ of its elements $x$, the brace notation ${x: P(x)}$ or ${x \mid P(x)}$ is also often used, and is read as ‘the set of all $x$ such that the statement $P(x)$ about $x$ is true.’ For example, $A={x: x$ is an even positive integer $<10}$.

## 数学代写|现代代数代写Modern Algebra代考|Relations on Sets

In mathematics, two types of very important relations, such as equivalence relation and ordered relations arise frequently. Sometimes, we need study decompositions of a non-empty set $X$ into disjoint subsets whose union is the entire set $X$ (i.e., $X$ is filled up by these subsets). Equivalence relations on $X$ provide tools to generate such decompositions of $X$ and produce new sets bearing a natural connection with the original set $X$.

A binary relation $R$ on a non-empty set $A$ is a mathematical concept and intuitively is a proposition such that for each ordered pair $(a, b)$ of elements of $A$, we can determine whether $a R b$ (read as $a$ is in relation $R$ to $b$ ) is true or false. We define it formally in terms of the set concept.

Definition 1.2.1 A binary relation $R$ on a non-empty set $A$ is a subset $R \subseteq A \times A$ and a binary relation $S$ from $A$ to $B$ is a subset $S$ of $A \times B$. The pair $(a, b) \in R$ is also denoted as $a R b$.

A binary relation $S$ from $A$ to $B$ is sometimes written as $S: A \rightarrow B$. Instead of writing a binary relation on $A$, we write only a relation on $A$, unless there is any confusion.

Example 1.2.1 For any set $A$, the diagonal $\Delta={(a, a): a \in A} \subseteq A \times A$ is the relation of equality.

In a binary relation $R$ on $A$, each pair of elements of $A$ need not be related i.e., $(a, b)$ may not belong to $R$ for all pairs $(a, b) \in A \times A$.
For example, if $a \neq b$, then $(a, b) \notin \Delta$ and also $(b, a) \notin \Delta$.
Example 1.2.2 The relation of inclusion on $\mathcal{P}(A)$ is ${(A, B) \in \mathcal{P}(A) \times \mathcal{P}(A): A \subseteq$ $B} \subseteq \mathcal{P}(A) \times \mathcal{P}(A)$

## 数学代写|现代代数代写Modern Algebra代考|Introductory Concepts

“集合”的概念在所有数学分支中都非常重要。我们遇到了一些不需要解释的术语或概念。此类术语称为末定义术 语并被视为原始概念。如果将术语“集合”定义为”集合是定义明确的对象集合”，那么集合的含义就不清楚了。可 以将“集合”定义为对象的“集合”。“聚合”是什么意思? 由于我们的语言是有限的，其他的同义词，如“阶级”、“家 庭”等，将会用尽。数学家接受有末定义的术语，“集合”应该是这样一个末定义的术语。但我们接受熟悉的表达方 式，例如“所有整数的集合”、“所有自然数的集合”、“所有有理数的集合”、“所有实数的集合”等。

## 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。