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# 数学代写|线性代数代写Linear algebra代考|MA2041 AFFINE COMBINATIONS

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

An affine combination of vectors is a special kind of linear combination. Given vectors (or “points”) $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_p$ in $\mathbb{R}^n$ and scalars $c_1, \ldots, c_p$, an affine combination of $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_p$ is a linear combination
$$c_1 \mathbf{v}_1+\cdots+c_p \mathbf{v}_p$$
such that the weights satisfy $c_1+\cdots+c_p=1$.

The set of all affine combinations of points in a set $S$ is called the affine hull (or affine span) of $S$, denoted by aff $S$.
The affine hull of a single point $\mathbf{v}_1$ is just the set $\left{\mathbf{v}_1\right}$, since it has the form $c_1 \mathbf{v}_1$ where $c_1=1$. The affine hull of two distinct points is often written in a special way. Suppose $\mathbf{y}=c_1 \mathbf{v}_1+c_2 \mathbf{v}_2$ with $c_1+c_2=1$. Write $t$ in place of $c_2$, so that $c_1=1-c_2=1-t$. Then the affine hull of $\left{\mathbf{v}_1, \mathbf{v}_2\right}$ is the set
$$\mathbf{y}=(1-t) \mathbf{v}_1+t \mathbf{v}_2, \quad \text { with } t \text { in } \mathbb{R}$$
This set of points includes $\mathbf{v}_1$ (when $t=0$ ) and $\mathbf{v}_2$ (when $t=1$ ). If $\mathbf{v}_2=\mathbf{v}_1$, then (1) again describes just one point. Otherwise, (1) describes the line through $\mathbf{v}_1$ and $\mathbf{v}_2$. To see this, rewrite (1) in the form
$$\mathbf{y}=\mathbf{v}_1+t\left(\mathbf{v}_2-\mathbf{v}_1\right)=\mathbf{p}+t \mathbf{u}, \quad \text { with } t \text { in } \mathbb{R}$$
where $\mathbf{p}$ is $\mathbf{v}_1$ and $\mathbf{u}$ is $\mathbf{v}_2-\mathbf{v}_1$. The set of all multiples of $\mathbf{u}$ is Span ${\mathbf{u}}$, the line through $\mathbf{u}$ and the origin. Adding $\mathbf{p}$ to each point on this line translates Span ${\mathbf{u}}$ into the line through p parallel to the line through $\mathbf{u}$ and the origin. See Figure 1. (Compare this figure with Figure 5 in Section 1.5.)

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

This section continues to explore the relation between linear concepts and affine concepts. Consider first a set of three vectors in $\mathbb{R}^3$, say $S=\left{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\right}$. If $S$ is linearly dependent, then one of the vectors is a linear combination of the other two vectors. What happens when one of the vectors is an affine combination of the others? For instance, suppose that
$$\mathbf{v}_3=(1-t) \mathbf{v}_1+t \mathbf{v}_2, \quad \text { for some } t \text { in } \mathbb{R} \text {. }$$
Then
$$(1-t) \mathbf{v}_1+t \mathbf{v}_2-\mathbf{v}_3=\mathbf{0} .$$
This is a linear dependence relation because not all the weights are zero. But more is true – the weights in the dependence relation sum to 0 :
$$(1-t)+t+(-1)=0 .$$
This is the additional property needed to define affine dependence.
An indexed set of points $\left{\mathbf{v}_1, \ldots, \mathbf{v}_p\right}$ in $\mathbb{R}^n$ is affinely dependent if there exist real numbers $c_1, \ldots, c_p$, not all zero, such that
$$c_1+\cdots+c_p=0 \text { and } c_1 \mathbf{v}_1+\cdots+c_p \mathbf{v}_p=\mathbf{0}$$
Otherwise, the set is affinely independent.

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

$$c_1 \mathbf{v}_1+\cdots+c_p \mathbf{v}_p$$

$$\mathbf{y}=(1-t) \mathbf{v}_1+t \mathbf{v}_2, \quad \text { with } t \text { in } \mathbb{R}$$

$$\mathbf{y}=\mathbf{v}_1+t\left(\mathbf{v}_2-\mathbf{v}_1\right)=\mathbf{p}+t \mathbf{u}, \quad \text { with } t \text { in } \mathbb{R}$$

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

\left 缺少或无法识别的分隔符 . 如果 $S$ 线性相关，则其中一个向量是其他两个向量的线性组 合。当其中一个向量是其他向量的仿射组合时会发生什么? 例如，假设
$$\mathbf{v}_3=(1-t) \mathbf{v}_1+t \mathbf{v}_2, \quad \text { for some } t \text { in } \mathbb{R} .$$

$$(1-t) \mathbf{v}_1+t \mathbf{v}_2-\mathbf{v}_3=\mathbf{0} .$$

$$(1-t)+t+(-1)=0 .$$

$$c_1+\cdots+c_p=0 \text { and } c_1 \mathbf{v}_1+\cdots+c_p \mathbf{v}_p=\mathbf{0}$$

## 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。