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• 数值分析
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• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|Quotient spaces

In order to motivate the introduction of the concept of quotient spaces, we first consider a concrete example in $\mathbb{R}^{2}$.
Let $v \in \mathbb{R}^{2}$ be any nonzero vector. Then
$$V=\operatorname{Span}{v}$$
represents the line passing through the origin and along (or opposite to) the direction of $v$. More generally, for any $u \in \mathbb{R}^{2}$, the coset
$$[u]={u+w \mid w \in V}=\left{x \in \mathbb{R}^{2} \mid x-u \in V\right}$$
represents the line passing through the vector $u$ and parallel to the vector $v$. Naturally, we define $\left[u_{1}\right]+\left[u_{2}\right]=\left{x+y \mid x \in\left[u_{1}\right], y \in\left[u_{2}\right]\right}$ and claim
$$\left[u_{1}\right]+\left[u_{2}\right]=\left[u_{1}+u_{2}\right] .$$
In fact, let $z \in\left[u_{1}\right]+\left[u_{2}\right]$. Then there exist $x \in\left[u_{1}\right]$ and $y \in\left[u_{2}\right]$ such that $z=x+y$. Rewrite $x, y$ as $x=u_{1}+w_{1}, y=u_{2}+w_{2}$ for some $w_{1}, w_{2} \in V$. Hence $z=\left(u_{1}+u_{2}\right)+\left(w_{1}+w_{2}\right)$, which implies $z \in\left[u_{1}+u_{2}\right]$. Conversely, if $z \in\left[u_{1}+u_{2}\right]$, then there is some $w \in V$ such that $z=\left(u_{1}+u_{2}\right)+w=$ $\left(u_{1}+w\right)+u_{2}$. Since $u_{1}+w \in\left[u_{1}\right]$ and $u_{2} \in\left[u_{2}\right]$, we see that $z \in\left[u_{1}\right]+\left[u_{2}\right]$. Hence the claim follows.

From the property (1.6.3), we see clearly that the coset $[0]=V$ serves as an additive zero element among the set of all cosets.

Similarly, we may also naturally define $a[u]={a x \mid x \in[u]}$ for $a \in \mathbb{R}$ where $a \neq 0$. Note that this last restriction is necessary because otherwise $0[u]$ would be a single-point set consisting of zero vector only. We claim
$$a[u]=[a u], \quad a \in \mathbb{R} \backslash{0} .$$

## 线性代数作业代写linear algebra代考|Normed spaces

It will be desirable to be able to evaluate the ‘length’ or ‘magnitude’ or ‘amplitude’ of any vector in a vector space. In other words, it will be useful to associate to each vector a quantity that resembles the notion of length of a vector in (say) $\mathbb{R}^{3}$. Such a quantity is generically called norm.
In this section, we take the field $\mathbb{F}$ to be either $\mathbb{R}$ or $\mathbb{C}$.
Definition 1.13 Let $U$ be a vector space over the field $\mathbb{F}$. A norm over $U$ is a correspondence $|\cdot|: U \rightarrow \mathbb{R}$ such that we have the following.
(1) (Positivity) $|u| \geq 0$ for $u \in U$ and $|u|=0$ only for $u=0$.
(2) (Homogeneity) $|a u|=|a||u|$ for $a \in \mathbb{F}$ and $u \in U$.
(3) (Triangle inequality) $|u+v| \leq|u|+|v|$ for $u, v \in U$.
A vector space equipped with a norm is called a normed space. If $|\cdot|$ is the specific norm of the normed space $U$, we sometimes spell this fact out by stating ‘normed space $(U,|\cdot|)$ ‘.

## 线性代数作业代写linear algebra代考|Quotient spaces

$$V=\operatorname{Span} v$$

$$[u]={u+w \backslash \text { mid } w \backslash \text { in } V}=\backslash \text { left }{x \backslash \text { in } \backslash \text { mathbb }{R} \wedge{2} \backslash \text { mid } x u \backslash \text { in } V \text { Vight }}$$

$$\left[u_{1}\right]+\left[u_{2}\right]=\left[u_{1}+u_{2}\right] .$$

$z=\left(u_{1}+u_{2}\right)+w=\left(u_{1}+w\right)+u_{2}$. 自从 $u_{1}+w \in\left[u_{1}\right]$ 和 $u_{2} \in\left[u_{2}\right]$, 我们看到 $z \in\left[u_{1}\right]+\left[u_{2}\right]$. 因此，索赔如下。

$$a[u]=[a u], \quad a \in \mathbb{R} \backslash 0 .$$

## 线性代数作业代写linear algebra代考|Normed spaces

(1) (积极性) $|u| \geq 0$ 为了 $u \in U$ 和 $|u|=0$ 仅适用于 $u=0$.
(2) (同质性) $|a u|=|a||u|$ 为了 $a \in \mathbb{F}$ 和 $u \in U$.
(3) (三角不等式) $|u+v| \leq|u|+|v|$ 为了 $u, v \in U$.

# 计量经济学代写

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions