# 线性代数网课代修|高阶线性代数代写Advanced Linear Algebra代考|Math471

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• 数值分析
• 高等线性代数
• 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|Dual spaces

Let $U$ be an $n$-dimensional vector space over a field $\mathbb{F}$. A functional (also called a form or a 1-form) $f$ over $U$ is a linear function $f: U \rightarrow \mathbb{F}$ satisfying $f(u+v)=f(u)+f(v), \quad u, v \in U ; \quad f(a u)=a f(u), \quad a \in \mathbb{F}, u \in U .$
Let $f, g$ be two functionals. Then we can define another functional called the sum of $f$ and $g$, denoted by $f+g$, by
$$(f+g)(u)=f(u)+g(u), \quad u \in U .$$
Similarly, let $f$ be a functional and $a \in \mathbb{F}$. We can define another functional called the scalar multiple of $a$ with $f$, denoted by $a f$, by
$$(a f)(u)=a f(u), \quad u \in U .$$
It is a simple exercise to check that these two operations make the set of all functionals over $U$ a vector space over $\mathbb{F}$. This vector space is called the dual space of $U$, denoted by $U^{\prime}$.

Let $\left{u_{1}, \ldots, u_{n}\right}$ be a basis of $U$. For any $f \in U^{\prime}$ and any $u=a_{1} u_{1}+\cdots+$ $a_{n} u_{n} \in U$, we have
$$f(u)=f\left(\sum_{i=1}^{n} a_{i} u_{i}\right)=\sum_{i=1}^{n} a_{i} f\left(u_{i}\right) .$$

## 线性代数作业代写linear algebra代考|Constructions of vector spaces

Let $U$ be a vector space and $V, W$ its subspaces. It is clear that $V \cap W$ is also a subspace of $U$ but $V \cup W$ in general may fail to be a subspace of $U$. The smallest subspace of $U$ that contains $V \cup W$ should contain all vectors in $U$ of the form $v+w$ where $v \in V$ and $w \in W$. Such an observation motivates the following definition.

Definition 1.9 If $U$ is a vector space and $V, W$ its subspaces, the sum of $V$ and $W$, denoted by $V+W$, is the subspace of $U$ given by
$$V+W \equiv{u \in U \mid u=v+w, v \in V, w \in W} .$$
Checking that $V+W$ is a subspace of $U$ that is also the smallest subspace of $U$ containing $V \cup W$ will be assigned as an exercise.

Now let $\mathcal{B}{0}=\left{u{1}, \ldots, u_{k}\right}$ be a basis of $V \cap W$. Expand it to obtain bases for $V$ and $W$, respectively, of the forms
$$\mathcal{B}{V}=\left{u{1}, \ldots, u_{k}, v_{1}, \ldots, v_{l}\right}, \quad \mathcal{B}{W}=\left{u{1}, \ldots, u_{k}, w_{1}, \ldots, w_{m}\right} .$$
From the definition of $V+W$, we get
$$V+W=\operatorname{Span}\left{u_{1}, \ldots, u_{k}, v_{1}, \ldots, v_{l}, w_{1}, \ldots, w_{m}\right}$$

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

$f: U \rightarrow \mathbb{F}$ 令人满意的
$f(u+v)=f(u)+f(v), \quad u, v \in U ; \quad f(a u)=a f(u), \quad a \in \mathbb{F}, u \in U .$

$$(f+g)(u)=f(u)+g(u), \quad u \in U$$

$$(a f)(u)=a f(u), \quad u \in U .$$

$$f(u)=f\left(\sum_{i=1}^{n} a_{i} u_{i}\right)=\sum_{i=1}^{n} a_{i} f\left(u_{i}\right)$$

## 线性代数作业代写linear algebra代考|Constructions of vector spaces

$$V+W \equiv u \in U \mid u=v+w, v \in V, w \in W .$$

# 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