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数学代考|线性代数代写Linear algebra代考|MA2210 Sums and Scalar Multiples

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数学代考|线性代数代写LINEAR ALGEBRA代考|Sums and Scalar Multiples

When a matrix $B$ multiplies a vector $\mathbf{x}$, it transforms $\mathbf{x}$ into the vector $B \mathbf{x}$. If this vector is then multiplied in turn by a matrix $A$, the resulting vector is $A(B \mathbf{x})$. See Figure 2 .
FIGURE 2 Multiplication by $B$ and then $A$.
Thus $A(B \mathbf{x})$ is produced from $\mathbf{x}$ by a composition of mappings – the linear transformations studied in Section 1.8. Our goal is to represent this composite mapping as multiplication by a single matrix, denoted by $A B$, so that
$$A(B \mathbf{x})=(A B) \mathbf{x}$$
See Figure $3 .$
FIGURE 3 Multiplication by $A B$.
If $A$ is $m \times n, B$ is $n \times p$, and $\mathbf{x}$ is in $\mathbb{R}^{p}$, denote the columns of $B$ by $\mathbf{b}{1}, \ldots, \mathbf{b}{p}$ and the entries in $\mathbf{x}$ by $x_{1}, \ldots, x_{p}$. Then
$$B \mathbf{x}=x_{1} \mathbf{b}{1}+\cdots+x{p} \mathbf{b}{p}$$ By the linearity of multiplication by $A$, \begin{aligned} A(B \mathbf{x}) &=A\left(x{1} \mathbf{b}{1}\right)+\cdots+A\left(x{p} \mathbf{b}{p}\right) \ &=x{1} A \mathbf{b}{1}+\cdots+x{p} A \mathbf{b}_{p} \end{aligned}

The vector $A(B \mathbf{x})$ is a linear combination of the vectors $A \mathbf{b}{1}, \ldots, A \mathbf{b}{p}$, using the entries in $\mathbf{x}$ as weights. In matrix notation, this linear combination is written as
$$A(B \mathbf{x})=\left[\begin{array}{llll} A \mathbf{b}{1} & A \mathbf{b}{2} & \cdots & A \mathbf{b}{p} \end{array}\right] \mathbf{x}$$ Thus multiplication by $\left[\begin{array}{llll}A \mathbf{b}{1} & A \mathbf{b}{2} & \cdots & A \mathbf{b}{p}\end{array}\right]$ transforms $\mathbf{x}$ into $A(B \mathbf{x})$. We have found the matrix we sought!

数学代考|线性代数代写LINEAR ALGEBRA代考|Properties of M

The following theorem lists the standard properties of matrix multiplication. Recall that $I_{m}$ represents the $m \times m$ identity matrix and $I_{m} \mathbf{x}=\mathbf{x}$ for all $\mathbf{x}$ in $\mathbb{R}^{m}$.
Let $A$ be an $m \times n$ matrix, and let $B$ and $C$ have sizes for which the indicated sums and products are defined.
a. $A(B C)=(A B) C \quad$ (associative law of multiplication)
b. $A(B+C)=A B+A C \quad$ (left distributive law)
c. $(B+C) A=B A+C A \quad$ (right distributive law)
d. $r(A B)=(r A) B=A(r B)$
for any scalar $r$
e. $I_{m} A=A=A I_{n}$
(identity for matrix multiplication)
PROOF Properties (b)-(e) are considered in the exercises. Property (a) follows from the fact that matrix multiplication corresponds to composition of linear transformations (which are functions), and it is known (or easy to check) that the composition of functions is associative. Here is another proof of (a) that rests on the “column definition” of the product of two matrices. Let
$$C=\left[\begin{array}{lll} \mathbf{c}{1} & \cdots & \mathbf{c}{p} \end{array}\right]$$
By the definition of matrix multiplication,
\begin{aligned} B C &=\left[\begin{array}{lll} B \mathbf{c}{1} & \cdots & B \mathbf{c}{p} \end{array}\right] \ A(B C) &=\left[\begin{array}{lll} A\left(B \mathbf{c}{1}\right) & \cdots & A\left(B \mathbf{c}{p}\right) \end{array}\right] \end{aligned}
Recall from equation (1) that the definition of $A B$ makes $A(B \mathbf{x})=(A B) \mathbf{x}$ for all $\mathbf{x}$, so
$$A(B C)=\left[\begin{array}{lll} (A B) \mathbf{c}{1} & \cdots & (A B) \mathbf{c}{p} \end{array}\right]=(A B) C$$

数学代考|线性代数代写LINEAR ALGEBRA代考|Sums and Scalar Multiples

$A B$, 以便
$$A(B \mathbf{x})=(A B) \mathbf{x}$$

$$B \mathbf{x}=x_{1} \mathbf{b} 1+\cdots+x p \mathbf{b} p$$

$$A(B \mathbf{x})=A(x 1 \mathbf{b} 1)+\cdots+A(x p \mathbf{b} p) \quad=x 1 A \mathbf{b} 1+\cdots+x p A \mathbf{b}{p}$$ 向量 $A(B \mathbf{x})$ 是向量的线甡组合 $A \mathbf{b} 1, \ldots, A \mathbf{b} p$ ，使用中的夆目 $\mathbf{x}$ 作为权重。在矩阵表示法中，这种线侏组合写成 $$A(B \mathbf{x})=\left[\begin{array}{llll} A \mathbf{b} 1 & A \mathbf{b} 2 & \cdots & A \mathbf{b} p] \mathbf{x} \end{array}\right.$$

数学代考|线性代数代写LINEAR ALGEBRA代考|Properties of M

(矩阵乘法的恒等式)

$$C=\left[\begin{array}{lll} \mathbf{c} 1 & \cdots & \mathbf{c} p \end{array}\right]$$

$$B C=\left[\begin{array}{lll} B \mathbf{c} 1 & \cdots & B \mathbf{c} p \end{array}\right] A(B C) \quad=\left[\begin{array}{llll} A(B \mathbf{c} 1) & \cdots & A(B \mathbf{c} p) \end{array}\right]$$

$$A(B C)=\left[\begin{array}{lll} (A B) \mathbf{c} 1 & \cdots & (A B) \mathbf{c} p \end{array}\right]=(A B) C$$

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。