## 数学代写|线性代数代写linear algebra代考|MAST10022

2023年1月2日

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

Matrix factorizations or decompositions play an important role in numerical methods of computational linear algebra. They help in speeding up algorithms used in linear algebra such as solving linear systems, inverting a matrix or computing its determinant. One such decomposition is the LU factorization. In this section, we will describe this factorization as well as when and to what it applies. The LU factorization expresses a square matrix as a product of a unit lower triangular matrix times an upper triangular matrix, where a unit triangular matrix has ones on the diagonal.
Example 2.48 Below we express a matrix $A=L U$, where $L$ is unit lower triangular and $U$ is upper triangular. Shortly, we will see the algorithm for performing this factorization.
$$\left[\begin{array}{rrr} 1 & -3 & 1 \ 2 & -8 & -1 \ -3 & 1 & 1 \end{array}\right]=\left[\begin{array}{rrr} 1 & 0 & 0 \ 2 & 1 & 0 \ -3 & 4 & 1 \end{array}\right]\left[\begin{array}{rrr} 1 & -3 & 1 \ 0 & -2 & -3 \ 0 & 0 & 16 \end{array}\right]$$
We will show that such a factorization is possible when one can row reduce a square matrix to an upper triangular matrix using only Type 3 elementary row operations, $a R_i+R_j$. To prove this we need a lemma the proof of which is left as an exercise. We also point out that this is not always possible. For example, for the matrix $\left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}\right]$ it is not possible (exercise).
Lemma $2.8$ The following statements about unit triangular matrices hold.

1. A finite product of unit lower (upper) triangular matrices is unit lower (upper) triangular.
2. The inverse of a unit lower (upper) triangular matrix is unit lower (upper) triangular.

Note first that the elementary matrix corresponding to an elementary row operation of the form $a R_i+R_j$ with $i<j$ is a unit lower triangular matrix (exercise). We can now prove the main result in this section.

## 数学代写|线性代数代写linear algebra代考|DEFINITION AND EXAMPLES

The first two chapters were more or less an introduction to the notion of a vector space by way of two examples: $\mathbb{R}^n$ and $M_{m n}$, i.e. Tuples and Matrices. The first two chapters were also meant to hone the computational skills needed in linear algebra. These skills will be employed quite liberally in what is to follow.

Now we are ready to introduce the general notion of a vector space. Technically speaking we are introducing real vector space, i.e. where the scalars are real numbers, but we could easily allow scalars to be the complex numbers or even an arbitrary field. We start, of course, with its definition which should look quite familiar at this point in the text.

Definition 3.1 A vector space is a set $V$, made up of objects called vectors, together with two operations:

1. Scalar Multiplication, in which a scalar (real number) is multiplied by a vector. We will denote scalar multiplication of a scalar a and a vector $v$ by av.

The 0 in Property 3 is called the zero vector and the $v$ in Property 4 is called the additive inverse of $u$ and is denoted symbolically as $-u$ (the use of the definite article the in front of 0 and $v$ will be justified at the end of this section).

We now give the four classical examples of a vector space which will appear over and over again in the remainder of the text. For this reason the reader should be sure to have a comfortable familiarity with these four examples. The first two examples should already be quite familiar. For each example, in order for the example to be complete, it is necessary to define what the set of vectors are as well as the two operations of scalar multiplication and vector addition.

# 线性代数代考

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

1. 单位下 (上) 三角矩阵的有限积是单位下 (上) 三角矩阵。
2. 单位下 (上) 三角矩阵的逆矩阵是单位下 (上) 三角矩阵。
首先注意对应于形式的基本行操作的基本矩阵 $a R_i+R_j$ 和 $i<j$ 是单位下三角矩阵 (练习) 。我们现 在可以证明本节的主要结果。

## 数学代写|线性代数代写linear algebra代考|DEFINITION AND EXAMPLES

$\mathbb{R}^n$ 和 $M_{m n}$ ，即元组和矩阵。前两章还旨在磨练线性代数所需的计算技能。这些技能在接下来的内容中大量使用。

1. 标量乘法，其中标量 (实数) 乘以向量。我们将表示标量 a 和向量的标量乘法 $v$ 的城市。
性质 3 中的 0 称为零向量， $v$ 在性质 4 中称为加法逆 $u$ 并 象征性地表示为 $-u$ (在 0 和前面使用定冠词 the $v$ 将在 本节末尾进行说明）。
我们现在给出向量空间的四个经典例子，它们将在本文 的其余部分反复出现。因此，读者应该确保熟悉这四个示例。前两个例子应该已经很熟悉了。对于每一个例 子，为了例子完整，需要定义向量的集合是什么，以及标量乘法和向量加法这两个运算。

## 有限元方法代写

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## MATLAB代写

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