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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

2.6.3 Establish if the set $X=\left{p(x) \in \mathbb{R}3[x] \mid p(-1)=0\right}$ is a subspace of $\mathbb{R}_3[x]$, where $p(-1)$ means the value of the polynomial calculated in $-1$. 2.6.4 Determine whether the set $X=\left{p(x) \in \mathbb{R}_2[x] \mid p(1)=-1\right}$ is a subspace of $\mathbb{R}_2[x]$ 2.6.5 Determine if the set $X={g: \mathbb{R} \rightarrow \mathbb{R} \mid g$ is continuous and differentiable in $x=2}$ is a subspace of the vector space of continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$. 2.6.6 Determine if the set $X={g: \mathbb{R} \rightarrow \mathbb{R} \mid g$ is continuous but not differentiable in $x=0}$ and is a subspace of the vector space of continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$. 2.6.7 Write, if possible, the set $S=\left{(x, y) \in \mathbb{R}^2 \mid x^2+x y-2 y^2=0\right}$ as a union of two subspaces of $\mathbb{R}^2$ and say if $S$ is a subspace of $\mathbb{R}^2$. 2.6.8 We call sequence of elements in $\mathbb{R}$ any function $s: \mathbb{N} \rightarrow \mathbb{R}$. If $s(n)=a_n$, the sequence is also indicated with $\left(a_n\right)$. On the set $\mathcal{S}{\mathbb{R}}$ of all sequences with elements in $\mathbb{R}$, we define the following operations:
$$\left(a_n\right)+\left(b_n\right)=\left(a_n+b_n\right), \quad k\left(a_n\right)=\left(k a_n\right)$$
for every $\left(a_n\right),\left(b_n\right) \in \mathcal{S}{\mathbb{R}}$ and $k \in \mathbb{R}$. Show that with these operations $\mathcal{S}{\mathbb{R}}$ is a vector space over $\mathbb{R}$.
2.6.9 Let $\mathcal{C}(\mathbb{R} ; \mathbb{R})$ be the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Consider the operation of sum of functions and the operation of product of any function by a real number defined as in Example 2.3.3. Show that with these operations $\mathcal{C}(\mathbb{R} ; \mathbb{R})$ is a vector space over $\mathbb{R}$.

## 数学代写|线性代数代写linear algebra代考|LINEAR COMBINATIONS AND GENERATORS

Every vector space $V \neq{0}$ contains infinitely many vectors; for if $V$ contains a vector $\mathbf{v}$, it immediately must also contain all its multiples, i.e. $\lambda \mathbf{v} \in V$ for each $\lambda \in \mathbb{R}$. Let us see an example to better understand this fact.

Consider the subspace $W={(x, a x) \mid x \in \mathbb{R}}$ in $\mathbb{R}^2$ discussed in the previous chapter. It is represented, in the Cartesian plane, by a line whose equation is $y=a x$. We can describe it, in an alternative way, as the set of all multiples of the vector $(1, a)$
$$W={x(1, a) \quad \mid x \in \mathbb{R}}$$

We say that the vector $(1, a)$ generates the subspace $W$ represented by the line $y=x$. The word “generate” is not accidental since, in fact, all vectors of the subspace $W$ are multiples of $(1, a)$. We also note that the choice of the vector $(1, a)$, as a generator of $W$, is arbitrary, we could as well have choosen any of its multiples, like $(2,2 a)$ or $\left(-\frac{3}{2},-\frac{3}{2} a\right)$

Graphically it is clear that if we know a point of a straight line (in the plane, but also in three-dimensional space) different from the origin, then we can immediately draw the line passing through it and the origin. We will see later that the fact of knowing the generators of a vector space allows us to determine it uniquely.

Now let us see another example. In $\mathbb{R}^2$, we consider the two vectors $(1,0)$ and $(0,1)$. We ask ourselves: what is the smallest subspace $W$ of $\mathbb{R}^2$ that contains both of these vectors? From the previous reasoning, we know that this subspace must contain the two subspaces $W_1$ and $W_2$ generated by $(1,0)$ and $(0,1)$ :
$W_1={\lambda(1,0) \mid \lambda \in \mathbb{K}} \quad$ represented by the $x$-axis
$W_2={\mu(0,1) \mid \mu \in \mathbb{R}} \quad$ represented by the $y$-axis
We also know that the sum of two vectors of $W$ still belongs to $W$ (by the definition of subspace $)$. For instance $(1,0)+(0,1)=(1,1) \in W$, but also $(1,2)+(3,4)=$ $(4,6) \in W$. The student is invited to draw vectors sums in $\mathbb{R}^2$ considering the points of the plan associated with them and using the parallelogram rule. In this way, we can convince ourselves that actually $W=\mathbb{R}^2$. But the graphic construction is not sufficient to prove this fact, as it is not possible draw all the vectors of the plane, so let us look at an algebraic proof. We take the generic vector $(\lambda, 0)$ in $W_1$ and the generic vector $(0, \mu)$ in $W_2$, and we take their sum: $(\lambda, 0)+(0, \mu)=(\lambda, \mu)$.

# 线性代数代考

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

$2.6 .3$ 建立如果集合〈left 的分隔符缺失或无法识别

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师