## 数学代写|代数学代写Algebra代考|MTH350

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## 数学代写|代数学代写Algebra代考|Summary and Review

In this chapter, we introduced the central objects that are studied in linear algebra: vectors, matrices, and linear transformations. We developed some basic ways of manipulating and combining these objects, such as vector addition and scalar multiplication, and we saw that these operations satisfy the basic properties, like distributivity and associativity, that we would expect them to based on our familiarity with properties of real numbers.

On the other hand, the formula for matrix multiplication was seemingly quite bizarre at first, but was later justified by the fact that it implements the action of linear transformations. That is, we can think of a linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ as being “essentially the same” as its standard matrix
$$[T]=\left[T\left(\mathbf{e}_1\right)\left|T\left(\mathbf{e}_2\right)\right| \cdots \mid T\left(\mathbf{e}_n\right)\right]$$
in the following two senses:

• Applying $T$ to $\mathrm{v}$ is equivalent to performing matrix-vector multiplication with $[T]$. That is, $T(\mathbf{v})=[T] \mathbf{v}$.
• Composing two linear transformations $S$ and $T$ is equivalent to multiplying their standard matrices. That is, $[S \circ T]=[S][T]$.
For these reasons, we often do not even differentiate between matrices and linear transformations in the later sections of this book. Instead, we just talk about matrices, with the understanding that a matrix is no longer “just” a 2D array of numbers for us, but is also a function that moves vectors around $\mathbb{R}^n$ in a linear way (i.e., it is a linear transformation). Furthermore, the columns of the matrix tell us exactly where the linear transformation sends the standard basis vectors $\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n$ (see Figure $1.22$ ).

This interpretation of matrices reinforces the idea that most linear algebraic objects have both an algebraic interpretation as well as a geometric one. Most importantly, we have the following interpretations of vectors and matrices:

• Algebraically, vectors are lists of numbers. Geometrically, they are arrows in space that represent movement or displacement.
• Algebraically, matrices are arrays of numbers. Geometrically, they are linear transformations – functions that deform a square grid in $\mathbb{R}^n$ into a parallelogram grid.

## 数学代写|代数学代写Algebra代考|Areas, Volumes, and the Cross Product

There is one more operation on vectors that we have not yet introduced, called the cross product. To help motivate it, consider the problem of finding a vector that is orthogonal to $\mathbf{v}=\left(v_1, v_2\right) \in \mathbb{R}^2$. It is clear from inspection that one vector that works is $\mathbf{w}=\left(v_2,-v_1\right)$, since then $\mathbf{v} \cdot \mathbf{w}=v_1 v_2-v_2 v_1=0$ (see Figure 1.23(a)).

If we ramp this type of problem up slightly to 3 dimensions, we can instead ask for a vector $\mathbf{x} \in \mathbb{R}^3$ that is orthogonal to two vectors $\mathbf{v}=\left(v_1, v_2, v_3\right) \in \mathbb{R}^3$ and $\mathbf{w}=\left(w_1, w_2, w_3\right) \in \mathbb{R}^3$. It is much more difficult to eyeball a solution in this case, but we will verify momentarily that the following vector works:
If $\mathbf{v}=\left(v_1, v_2, v_3\right) \in \mathbb{R}^3$ and $\mathbf{w}=\left(w_1, w_2, w_3\right) \in \mathbb{R}^3$ are vectors then their cross product, denoted by $\mathbf{v} \times \mathbf{w}$, is defined by
$$\mathbf{v} \times \mathbf{w}=\left(\begin{array}{l} v_2 w_3-v_3 w_2 \ v_3 w_1-v_1 w_3 \ v_1 w_2-v_2 w_1 \end{array}\right)$$
To see that the cross product is orthogonal to each of $\mathbf{v}$ and $\mathbf{w}$, we simply compute the relevant dot products:
\begin{aligned} \mathbf{v} \cdot(\mathbf{v} \times \mathbf{w}) &=v_1\left(v_2 w_3-v_3 w_2\right)+v_2\left(v_3 w_1-v_1 w_3\right)+v_3\left(v_1 w_2-v_2 w_1\right) \ &=v_1 v_2 w_3-v_1 v_3 w_2+v_2 v_3 w_1-v_2 v_1 w_3+v_3 v_1 w_2-v_3 v_2 w_1 \ &=0 . \end{aligned}
The dot product $\mathbf{w} \cdot(\mathbf{v} \times \mathbf{w})$ can similarly be shown to equal 0 (see Exercise 1.A.10), so we conclude that $\mathbf{v} \times \mathbf{w}$ is orthogonal to each of $\mathbf{v}$ and $\mathbf{w}$, as illustrated in Figure $1.23$ (b).

# 代数学代写

## 数学代写|代数学代写Algebra代考|Summary and Review

. .

$$[T]=\left[T\left(\mathbf{e}_1\right)\left|T\left(\mathbf{e}_2\right)\right| \cdots \mid T\left(\mathbf{e}_n\right)\right]$$
“本质相同”，具有以下两种意义:

• 正在应用 $T$ 到 $\mathrm{v}$ 等价于用 $[T]$。也就是说， $T(\mathbf{v})=[T] \mathbf{v}$.
• 构成两个线性变换 $S$ 和 $T$ 等于乘以它们的标准矩阵。也就是说， $[S \circ T]=[S][T]$由于这些原因，在本书后面的章节中，我们通常甚至不区分矩阵和线性变换。相反，我们只讨论矩阵，理解矩阵对我们来说不再“只是”一个二维数字数组，它还是一个移动向量的函数 $\mathbb{R}^n$ 以线性的方式(即，它是一个线性变换)。此外，矩阵的列准确地告诉我们线性变换将标准基向量发送到哪里 $\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n$ (见图 $1.22$

## 数学代写|代数学代写Algebra代考|区域，卷，和叉乘

$$\mathbf{v} \times \mathbf{w}=\left(\begin{array}{l} v_2 w_3-v_3 w_2 \ v_3 w_1-v_1 w_3 \ v_1 w_2-v_2 w_1 \end{array}\right)$$

\begin{aligned} \mathbf{v} \cdot(\mathbf{v} \times \mathbf{w}) &=v_1\left(v_2 w_3-v_3 w_2\right)+v_2\left(v_3 w_1-v_1 w_3\right)+v_3\left(v_1 w_2-v_2 w_1\right) \ &=v_1 v_2 w_3-v_1 v_3 w_2+v_2 v_3 w_1-v_2 v_1 w_3+v_3 v_1 w_2-v_3 v_2 w_1 \ &=0 . \end{aligned}
$\mathbf{w} \cdot(\mathbf{v} \times \mathbf{w})$同样可以被表示为等于0(参见练习1.A.10)，因此我们得出结论:$\mathbf{v} \times \mathbf{w}$与$\mathbf{v}$和$\mathbf{w}$彼此正交，如图$1.23$ (b)所示

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。