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# 数学代写|线性代数代写Linear algebra代考|MATH307 Matrices

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## 数学代写|线性代数代写Linear algebra代考|Matrices

We will use matrices to develop systematic methods to solve linear systems and to study the properties of the solution set of a linear system. Informally speaking, a matrix is an array or table consisting of rows and columns. For example,
$$\mathbf{A}=\left[\begin{array}{cccc} 1 & -2 & 1 & 0 \ 0 & 2 & -8 & 8 \ -4 & 7 & 11 & -5 \end{array}\right]$$
is a matrix having $m=3$ rows and $n=4$ columns. In general, a matrix with $m$ rows and $n$ columns is a $m \times n$ matrix and the set of all such matrices will be denoted by $M_{m \times n}$. Hence, $\mathbf{A}$ above is a $3 \times 4$ matrix. The entry of $\mathbf{A}$ in the $i$ th row and $j$ th column will be denoted by $a_{i j}$. A matrix containing only one column is called a column vector and a matrix containing only one row is called a row vector. For example, here is a row vector
$$\mathbf{u}=\left[\begin{array}{lll} 1 & -3 & 4 \end{array}\right]$$
and here is a column vector
$$\mathbf{v}=\left[\begin{array}{c} 3 \ -1 \end{array}\right]$$
We can associate to a linear system three matrices: (1) the coefficient matrix, (2) the output column vector, and (3) the augmented matrix. For example, for the linear system
\begin{aligned} 5 x_1-3 x_2+8 x_3 & =-1 \ x_1+4 x_2-6 x_3 & =0 \ 2 x_2+4 x_3 & =3 \end{aligned}
the coefficient matrix $\mathbf{A}$, the output vector $\mathbf{b}$, and the augmented matrix $[\mathbf{A} \mathbf{b}]$ are:
$$\mathbf{A}=\left[\begin{array}{ccc} 5 & -3 & 8 \ 1 & 4 & -6 \ 0 & 2 & 4 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} -1 \ 0 \ 3 \end{array}\right], \quad[\mathbf{A} \mathbf{b}]=\left[\begin{array}{cccc} 5 & -3 & 8 & -1 \ 1 & 4 & -6 & 0 \ 0 & 2 & 4 & 3 \end{array}\right]$$

## 数学代写|线性代数代写Linear algebra代考|Solving linear systems

In algebra, you learned to solve equations by first “simplifying” them using operations that do not alter the solution set. For example, to solve $2 x=8-2 x$ we can add to both sides $2 x$ and obtain $4 x=8$ and then multiply both sides by $\frac{1}{4}$ yielding $x=2$. We can do similar operations on a linear system. There are three basic operations, called elementary operations, that can be performed:

Interchange two equations.

Multiply an equation by a nonzero constant.

Add a multiple of one equation to another.

These operations do not alter the solution set. The idea is to apply these operations iteratively to simplify the linear system to a point where one can easily write down the solution set. It is convenient to apply elementary operations on the augmented matrix [A b] representing the linear system. In this case, we call the operations elementary row operations, and the process of simplifying the linear system using these operations is called row reduction. The goal with row reducing is to transform the original linear system into one having a triangular structure and then perform back substitution to solve the system. This is best explained via an example.

Example 1.5. Use back substitution on the augmented matrix
$$\left[\begin{array}{cccc} 1 & 0 & -2 & -4 \ 0 & 1 & -1 & 0 \ 0 & 0 & 1 & 1 \end{array}\right]$$
to solve the associated linear system.
Solution. Notice that the augmented matrix has a triangular structure. The third row corresponds to the equation $x_3=1$. The second row corresponds to the equation
$$x_2-x_3=0$$
and therefore $x_2=x_3=1$. The first row corresponds to the equation
$$x_1-2 x_3=-4$$
and therefore
$$x_1=-4+2 x_3=-4+2=-2 .$$
Therefore, the solution is $(-2,1,1)$.

## 数学代写线性代数代写Linear algebra代考|Matrices

$$\mathbf{u}=\left[\begin{array}{lll} 1 & -3 & 4 \end{array}\right]$$

$$\mathbf{v}=\left[\begin{array}{ll} 3 & -1 \end{array}\right]$$

$$5 x_1-3 x_2+8 x_3=-1 x_1+4 x_2-6 x_3 \quad=02 x_2+4 x_3=3$$

## 数学代写|线性代数代写Linear algebra代考|Solving linear systems

$$x_2-x_3=0$$

$$x_1-2 x_3=-4$$

$$x_1=-4+2 x_3=-4+2=-2 .$$

## 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。