# 线性代数网课代修|张量代数代写Tensor algebra辅导|MATH2022

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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|Definition of relative tensors

In some books on tensors, the notion of tensor density or pseudotensor is discussed. In this section, after extending the concept of tensors, we give the definition of these concepts. In fact, we define a new concept which contains our tensor spaces and call it a generalized tensor space. Tensor density and pseudotensor are defined as special cases of it. Let $V$ be a vector space. Tensor density and pseudotensor are correspondences which associate a tensor to each basis for $V$ and which satisfy certain conditions. The conditions are given as follows: the relations between tensors associated to different bases are described by relations between corresponding bases. To explain the situation precisely, we use the notions of a set on which a group operates and of an equivariant mapping between these sets.

First, we shall define the notion of a group acting on a set. Let $G$ be a group. The law of composition on $G$ is denoted by $G \times G \ni\left(g, g^{\prime}\right) \mapsto g g^{\prime} \in$ $G$. We denote by $e$ the unit element of $G$ and by $g^{-1}$ the inverse of $g$. DEFinition 2.6. Let $X$ be a set. If there exists a mapping $X \times G \rightarrow X$, $(x, g) \mapsto x \cdot g$, which satisfies the following conditions (1) and (2), then we say that $G$ operates on $X$ on the right. (1) For all $g, g^{\prime} \in G$ and $x \in X, x \cdot\left(g g^{\prime}\right)=(x \cdot g) \cdot g^{\prime}$. (2) For all $x \in X, x \cdot e=x$. On the other hand, if there exists a mapping $G \times X \rightarrow X,(g, x) \mapsto g \cdot x$, which satisfies the following conditions $(1)^{\prime}$ and $(2)^{\prime}$, then we say that $G$ operates on $X$ on the left. $(1)^{\prime}$ For all $g, g^{\prime} \in G$ and $x \in X,\left(g g^{\prime}\right) \cdot x=g \cdot\left(g^{\prime} \cdot x\right)$. (2) For all $x \in X, e \cdot x=x$. EXAMPLE $2.20$. Let $V$ be a $k$-vector space of dimension $n$. Let $\mathscr{F}(V)$ be the set of all bases of $V$. Let $\mathscr{E}=\left(e_{1}, \ldots, e_{n}\right) \in \mathscr{F}(V)$ and $A=$ $\left(\alpha^{i}{ }{j}\right)\left(\alpha^{i}{ }{j} \in k\right)$ be a regular $n \times n$ matrix. Define $e_{j}^{\prime} \in V$ by $e_{j}^{\prime}=\sum_{i=1}^{n} \alpha^{i}{ }{j} e{i}$ $(j=1, \ldots, n)$. Then $\mathscr{E}^{\prime}=\left(e_{1}^{\prime}, \ldots, e_{n}^{\prime}\right)$ is also an element of $\mathscr{F}(V)$. If we define $\mathscr{E}^{\prime}=\mathscr{E} \cdot A$, the group $\mathrm{GL}_{n}(k)$ of regular $n \times n$ matrices over $k$ operates on $\mathscr{F}(V)$ on the right.

## 线性代数作业代写linear algebra代考|Definition of exterior algebra and its properties

We denote by $A^{p}(V)$ the space of all alternating tensors of degree $p$. Recall that $A^{1}(V)=T^{1}(V)=V$ and $A^{m}(V)={0}$ for $m>n=\operatorname{dim} V$ (see $\S 2.3(\mathrm{~b}))$. Let $A(V)$ be the direct sum of $A^{p}(V)$, where $A^{0}(V)=T^{0}(V)=k$. $$A(V)=\bigoplus_{p=0}^{\infty} A^{p}(V)=\bigoplus_{p=0}^{n} A^{p}(V)$$ Next, we define a multiplication on $A(V)$. Let $t \in A^{p}(V)$ and $t^{\prime} \in A^{q}(V)$. Then, $t \otimes t^{\prime}$ is an element of $T^{p+q}(V)$ and we can apply the alternator $\mathscr{A}{p+q}$. The element $\mathscr{A}{p+q}\left(t \otimes t^{\prime}\right)$ is called the exterior product of $t$ and $t^{\prime}$ and is denoted by $t \wedge t^{\prime}$. If $p+q>n$, the alternator $\mathscr{A}{p+q}$ is the zero mapping. This product is extended to all elements of $A(V)$ by bilinearlity. Namely, for $t=\sum t{p}, t^{\prime}=\sum t_{p}^{\prime}\left(t_{p}, t_{p}^{\prime} \in A^{p}(V)\right)$ of $A(V), t \wedge t^{\prime}$ is defined by $$t \wedge t^{\prime}=\sum_{p, q}\left(t_{p} \wedge t_{q}\right)=\sum_{k=0}^{n} \mathscr{A}{k}\left(\sum{p+q=k} t_{p} \otimes t_{q}^{\prime}\right)$$ EXAMPLE 3.1. Let $v, v^{\prime} \in V=A^{\prime}(V)$. Then, $$v \wedge v^{\prime}=\mathscr{A}_{2}\left(v \otimes v^{\prime}\right)=\frac{1}{2}\left(v \otimes v^{\prime}-v^{\prime} \otimes v\right)$$

## 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