# 线性代数网课代修|张量代数代写Tensor algebra辅导|MATH489

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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|Examples of tensor product spaces

(a) The space of linear mappings. Let $V$ and $W$ be $k$-vector spaces. The $k$-vector space $\operatorname{Hom}(V, W)$ consisting of all linear mappings of $V$ into $W$ can be regarded as a tensor product space.

Let $\varphi \in V^{}, w \in W$. For $(\varphi, w) \in V^{} \times W$, we define an element $F_{\varphi, w} \in \operatorname{Hom}(V, W)$ as follows:
$$F_{\varphi, w}(v)=\varphi(v) w \quad(v \in V) .$$
It is easy to see that $F_{\varphi, w}$ is an element of $\operatorname{Hom}(V, W)$.
Proposition 1.11. The correspondence $\left(\varphi \otimes w \leftrightarrow F_{\varphi, w}\right)$, gives an isomorphism between $V^{*} \otimes W$ and $\operatorname{Hom}(V, W)$.

Proof. The mapping $\Phi: V^{} \times W \rightarrow \operatorname{Hom}(V, W)$, which assigns $F_{\varphi, w}$ to $(\varphi, w) \in V^{} \times W$, is clearly bilinear. Therefore, by condition (T2) applied to $V^{} \otimes W$, there exists a linear mapping $\tilde{F}: V^{} \otimes W \rightarrow \operatorname{Hom}(V, W)$ such that $\tilde{F} \circ l=\Phi$.

## 线性代数作业代写linear algebra代考|Construction of tensor products with generators and relations

In this section we shall examine a third method of constructing tensor products. Though this method is not so easy to understand, it is theoretically interesting, since it can be widely applied.

First, we describe a $k$-vector space generated by a set $A$, where $A$ is a nonempty arbitrary (infinite or finite) set. Consider a $k$-vector space $\mathscr{V}(A)$ which has a basis ${e(a) \mid a \in A}$ such that, between the elements of $A$ and the basis elements, there is a bijective correspondence $e(a) \leftrightarrow a \quad(a \in A)$. $\operatorname{dim} \mathscr{V}(A)$ is equal to the cardinality of $A$, therefore, it is not necessarily finite. $\mathscr{V}(A)$ is called a vector space generated by $A$. Namely, $\mathscr{V}(A)$ is the set of all finite linear combinations of $e(a)(a \in A)$ with coefficients in $k$,
$$\sum_{a \in A} \alpha(a) e(a) \quad(\alpha(a) \in k) .$$
Here, “finite” means that the number of elements $a \in A$ with $\alpha(a) \neq 0$ is finite. Two linear combinations $\sum_{a \in A} \alpha(a) e(a)$ and $\sum_{a \in A} \beta(a) e(a)$ are equal if and only if $\alpha(a)=\beta(a)$ for every $a \in A$. Addition and the scalar multiplication are defined as follows:
$$\begin{gathered} \sum_{a \in A} \alpha(a) e(a)+\sum_{a \in A} \beta(a) e(a)=\sum_{a \in A}(\alpha(a)+\beta(a)) e(a) \ \gamma \cdot \sum_{a \in A} \alpha(a) e(a)=\sum_{a \in A}(\gamma \alpha(a)) e(a) \quad(\gamma \in k) \end{gathered}$$

## 线性代数作业代写linear algebra代考|Examples of tensor product spaces

(a) 线性映射空间。让 $V$ 和 $W$ 是 $k$-向量空间。这 $k$-向量空间 $\operatorname{Hom}(V, W)$ 由所有线性映射组 成 $V$ 进入 $W$ 可以看成一个张量积空间。

$$F_{\varphi, w}(v)=\varphi(v) w \quad(v \in V) .$$

## 线性代数作业代写linear algebra代考|Construction of tensor products with generators and relations

$$\sum_{a \in A} \alpha(a) e(a) \quad(\alpha(a) \in k)$$

$$\sum_{a \in A} \alpha(a) e(a)+\sum_{a \in A} \beta(a) e(a)=\sum_{a \in A}(\alpha(a)+\beta(a)) e(a) \gamma \cdot \sum_{a \in A} \alpha(a) e(a)=\sum_{a \in A}(\gamma \alpha(a)) e(a) \quad(\gamma \in k)$$

# 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