# 线性代数网课代修|张量代数代写Tensor algebra辅导|Math403

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## 线性代数作业代写linear algebra代考|Definition of tensor products

We are now ready to define the tensor product of $k$-vector spaces.
Definition 1.2. Let $V$ and $W$ be $k$-vector spaces. The pair $\left(U_{0}, l\right)$ consisting of a $k$-vector space $U_{0}$ and a bilinear mapping $l: V \times W \rightarrow U_{0}$, satisfying (T1) and (T2), the existence of which is assured by Theorem 1.1, is called a tensor product of $V$ and $W$. We write $U_{0}=V \otimes W$ and $t(v, w)=$ $v \otimes w$. The mapping $l$ is called the canonical mapping of a tensor product $U_{0}=V \otimes W$.

In the following, we sometimes say that a vector space $U_{0}$ is a tensor product $V \otimes W$ of $V$ and $W$. Implicitly this means that there exists a bilinear mapping $t: V \times W \rightarrow U_{0}$ satisfying (T1) and (T2) (or (T)). In this notation, condition (T) can be restated as follows: a tensor product $U_{0}$ of $V$ and $W$ is generated by ${v \otimes w(=l(v, w)) \mid v \in V, w \in W}$. The uniqueness property (2) of Theorem $1.1$ can be restated as follows: if $U_{0}$ and $U_{0}^{\prime}$ are tensor products of $V$ and $W$, then there exists a unique linear isomorphism $F_{0}: U_{0} \rightarrow U_{0}^{\prime}$ such that $F_{0}$ associates $v \otimes w$ in $U_{0}^{\prime}$ to $v \otimes w$ $(=l(v, w))$ in $U_{0}$ for all $v \in V, w \in W$.

In the proof of existence in Theorem 1.1, we used bases for $V$ and $W$. Therefore, it might be difficult to understand the meaning of the tensor product. Thus, we give another construction of a tensor product $\left(U_{0}, t\right)$ free from bases.

Let $V^{}$ and $W^{}$ be the dual spaces of $V$ and $W$ respectively (see $\S 1$ for dual spaces), and let $U_{0}$ be defined by
$$U_{0}=\mathscr{L}\left(V^{}, W^{} ; k\right) .$$

## 线性代数作业代写linear algebra代考|Properties of tensor products

We begin with properties which can be easily obtained. Rewriting the bilinearity of the canonical mapping $l$, we have the following proposition.
Proposition $1.1$ (bilinearity of $\otimes$ ). For $\alpha, \beta \in k, v, v_{1}, v_{2} \in V$, $w, w_{1}, w_{2} \in W$, we have
\begin{aligned} \left(\alpha v_{1}+\beta v_{2}\right) \otimes w &=\alpha\left(v_{1} \otimes w\right)+\beta\left(v_{2} \otimes w\right), \ v \otimes\left(\alpha w_{1}+\beta w_{2}\right) &=\alpha\left(v \otimes w_{1}\right)+\beta\left(v \otimes w_{2}\right) . \end{aligned}
From the proof of existence of tensor products $(\S 3)$ we obtain
Proposition 1.2. Let $\left(e_{1}, \ldots, e_{n}\right)$ be a basis for $V$ and $\left(f_{1}, \ldots, f_{m}\right) a$ basis for $W$. Then the mn elements $e_{i} \otimes f_{j}(1 \leq i \leq n, 1 \leq j \leq m)$ form a basis for $V \otimes W$.
In particular,
$$\operatorname{dim}(V \otimes W)=\operatorname{dim} V \cdot \operatorname{dim} W .$$
CoROLLARY. Let $v \in V$ and $w \in W$ be nonzero elements. Then $v \otimes w \neq$ 0 .

PRoof. Take bases of $V$ and $W$ which contain $v$ and $w$ respectively. Then $v \otimes w$ is a basis element of $V \otimes W$. Therefore $v \otimes w \neq 0$.

If we arrange the basis elements $e_{i} \otimes f_{j}$ in the order $e_{1} \otimes f_{1}, \ldots, e_{1} \otimes f_{m}$, $e_{2} \otimes f_{1}, \ldots, e_{2} \otimes f_{m}, \ldots, e_{n} \otimes f_{1}, \ldots, e_{n} \otimes f_{m}$, then we say that the basis elements are arranged in lexicographical order.

## 线性代数作业代写linear algebra代考|Definition of tensor products

$$U_{0}=\mathscr{L}(V, W ; k) .$$

## 线性代数作业代写linear algebra代考|Properties of tensor products

$$\left(\alpha v_{1}+\beta v_{2}\right) \otimes w=\alpha\left(v_{1} \otimes w\right)+\beta\left(v_{2} \otimes w\right), v \otimes\left(\alpha w_{1}+\beta w_{2}\right) \quad=\alpha\left(v \otimes w_{1}\right)+\beta\left(v \otimes w_{2}\right) .$$

$$\operatorname{dim}(V \otimes W)=\operatorname{dim} V \cdot \operatorname{dim} W .$$

$e_{2} \otimes f_{1}, \ldots, e_{2} \otimes f_{m}, \ldots, e_{n} \otimes f_{1}, \ldots, e_{n} \otimes f_{m}$ ，那么我们说基本元素是按字典顺序排列的。

# 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