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

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## 数学代写|线性代数代写linear algebra代考|APPLICATION: 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|BASIS AND DIMENSION

This section introduces what we call a counting principle for comparing the relative sizes of vector spaces, called dimension. It will conform to our intuition of dimension for $\mathbb{R}^2$ (2-dimensional) and $\mathbb{R}^3$ (3-dimensional), but in addition it will assign dimension to many other vector spaces. Our focus will be on investigating vector spaces of finite dimension.

Definition $3.12$ A set of vectors $v_1, \ldots, v_n \in V$ a vector space is a basis for $V$ if

1. The vectors $v_1, \ldots, v_n$ span $V$, and
2. The vectors $v_1, \ldots, v_n$ are linearly independent.
Example 3.40 Take the earlier example. We have already verified that $[1,0,0]$, $[1,1,0],[1,1,1]$ span $\mathbb{R}^3$. Now we show they are linearly independent (and thus a basis for $\mathbb{R}^3$ ). Again, we use the technique from the Section 3.3. Putting the vectors in columns in a determinant,

$$\left|\begin{array}{lll} 1 & 1 & 1 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{array}\right|=(1)(1)(1)=1 \neq 0$$
Example 3.41 The necessary generators of a particular span are a basis for that span. We shall formally prove this fact in Section 3.6. However, we illustrate this fact with one of the earlier examples. Although $[1,0,0],[1,1,0],[0,-2,0]$ do not span $\mathbb{R}^3$, from our calculations above, we know that $[1,0,0],[0,1,0]$ form a basis for $\operatorname{span}([1,0,0],[1,1,0],[0,-2,0])$, since $[1,0,0],[0,1,0]$ are the necessary generators of $\operatorname{span}([1,0,0],[1,1,0],[0,-2,0])$ and one can check that they are linearly independent.
We will not give too many examples at this point, because within this section we will shortcut the method of determining basis even further.

Definition 3.13 The following bases for their respective vector spaces are called standard bases:

1. The standard basis for $\mathbb{R}^n$ is the collection of vectors
$$e_1=[1,0, \ldots, 0], e_2=[0,1,0, \ldots, 0], \ldots, e_n=[0, \ldots, 0,1] .$$
Note that in $\mathbb{R}^2$ the notation for the standard basis is $\hat{\imath}, \hat{\jmath}$ and in $\mathbb{R}^3$ the notation is $\hat{\imath}, \hat{\jmath}, \hat{k}$
2. The standard basis for $P_n$ is the collection of vectors $1, x, x^2, \ldots, x^n$.
3. The standard basis for $P$ is the infinite collection of vectors $1, x, x^2, \ldots$.
4. The standard basis for $M_{m n}$ is the collection of vectors
$$\left{E_{i j} \mid 1 \leq i \leq m, 1 \leq j \leq n\right}$$
where each $E_{i j}$ is a matrix filled with zeros except that there is a 1 in the ijth entry.

# 线性代数代考

## 数学代写|线性代数代写线性代数代考|应用程序:LU FACTORIZATION

$$\left[\begin{array}{rrr} 1 & -3 & 1 \ 2 & -8 & -1 \ -3 & 1 & 1 \end{array}\right]=\left[\begin{array}{rrr} 1 & 0 & 0 \ 2 & 1 & 0 \ -3 & 4 & 1 \end{array}\right]\left[\begin{array}{rrr} 1 & -3 & 1 \ 0 & -2 & -3 \ 0 & 0 & 16 \end{array}\right]$$

## 数学代写|线性代数代写线性代数代考|BASIS AND DIMENSION

1. 向量$v_1, \ldots, v_n$张成$V$，
2. 向量$v_1, \ldots, v_n$是线性无关的。
例3.40举前面的例子。我们已经验证了$[1,0,0]$, $[1,1,0],[1,1,1]$ span $\mathbb{R}^3$。现在我们证明它们是线性无关的(因此是$\mathbb{R}^3$的基)。我们再次使用3.3节中的技术。将向量放在行列式中的列中，

$$\left|\begin{array}{lll} 1 & 1 & 1 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{array}\right|=(1)(1)(1)=1 \neq 0$$

$\mathbb{R}^n$的标准基是向量的集合
$$e_1=[1,0, \ldots, 0], e_2=[0,1,0, \ldots, 0], \ldots, e_n=[0, \ldots, 0,1] .$$

$$\left{E_{i j} \mid 1 \leq i \leq m, 1 \leq j \leq n\right}$$

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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。

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