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

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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|Properties of tensor spaces

(a) Components of tensors and their transformation laws. Let $V$ be a $k$ vector space and let $V^{}$ be the dual space of $V$. In this section we describe the properties of the tensor space $$T_{q}^{p}(V)=\underbrace{V \otimes \cdots \otimes V}{p \text { factors }} \otimes \underbrace{V^{} \otimes \cdots \otimes V^{*}}{q \text { factors }}$$
defined in $\S 1$.
If $\operatorname{dim}{k} V=n, \operatorname{dim}{k} T_{q}^{p}(V)=n^{p+q}$ since the dimension of a tensor product is the product of the dimensions of the factors. Choose a basis $\mathscr{E}=\left(e_{1}, \ldots, e_{n}\right)$ of $V$. Let $\left(f^{1}, \ldots, f^{n}\right)$ be the dual basis of $\mathscr{E}$. Then, the following $n^{p+q}$ elements
$$\left{e_{i_{1}} \otimes \cdots \otimes e_{i_{p}} \otimes f^{j_{1}} \otimes \cdots \otimes f^{j_{q}} \mid 1 \leq i_{\nu} \leq n, 1 \leq j_{\mu} \leq n\right}$$
form a basis for $T_{q}{ }^{p}(V)$ (Proposition $1.2^{\prime}$ ). We call this basis the standard basis for $T_{q}^{p}(V)$ corresponding to $\mathscr{E}$. To simplify the notation, set
$$e_{i_{1}} \otimes \cdots \otimes e_{i_{p}} \otimes f^{j_{1}} \otimes \cdots \otimes f^{j_{q}}=t_{i_{1}, \ldots, i_{p}}^{j_{1}, \cdots, j_{q}} .$$
Then an element $z$ of $T_{q}^{p}(V)$ can be uniquely expressed as
$$z=\sum_{i_{1}, \ldots, i_{p}, j_{1}, \ldots, j_{q}} \xi_{j_{1}, \ldots, j_{q}}^{i_{1}, \ldots, i_{p}} t_{i_{1}, \ldots, i_{p}}^{j_{1}, \ldots, j_{q}} \quad\left(\xi_{j_{1}, \ldots, j_{q}}^{i_{1}, \ldots, i_{p}} \in k\right) .$$
The $n^{p+q}$ elements $\left{\xi_{j_{1}, \ldots, j_{q}}^{i_{1}, \ldots, i_{p}}\right} \quad\left(1 \leq i_{\nu} \leq n, 1 \leq j_{\mu} \leq n\right)$ of $k$ (or the array of these elements) are called the components of $z$ with respect to the basis $\mathscr{E}$. (Precisely speaking, they should be called the components with respect to the standard basis corresponding to $\mathscr{E}$.)

## 线性代数作业代写linear algebra代考|Symmetric tensors and alternating tensors

(a) Definitions. There are families of tensors which are called symmetric or alternating. In this section, we give their definitions and study their properties. First, we consider the space $T^{p}(V)$ of contravariant tensors of degree $p$. In this section, we write $T^{p}(V)=T^{p}$ for simplicity.

Let $\mathfrak{S}{p}$ be the set of permutations of the set ${1, \ldots, p}$ with $p$ elements. Denote by $\operatorname{sgn}(\sigma)$ the signature of $\sigma \in \mathfrak{S}{p}$ (i.e., $\operatorname{sgn}(\sigma)=1$ if $\sigma$ is an even permutation and $\operatorname{sgn}(\sigma)=-1$ if $\sigma$ is an odd permutation.)

From the next proposition we know that there corresponds to every element $\sigma$ of $\mathfrak{S}{p}$ a linear transformation $P{\sigma}$ of $T^{p}$.

PRoposition 2.4. (1) Let $\sigma \in \mathfrak{S}{p}$. There exists a unique linear transformation $P{\sigma}$ of $T^{p}$ such that
$$P_{\sigma}\left(v_{1} \otimes \cdots \otimes v_{p}\right)=v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(p)} \quad\left(v_{i} \in V\right) .$$
Moreover, $P_{\sigma}$ is a linear isomorphism.
(2) For $\sigma, \tau \in \mathfrak{S}{p}, P{\sigma} P_{\tau}=P_{\sigma \tau}$.
Denote by 1 the identity permutation. Then
$$P_{1}=I \quad\left(=\text { the identity transformation of } T^{p}\right) .$$
Proof. (1) To prove the existence and the uniqueness of $P_{\sigma}$, consider a $p$-multilinear mapping defined as follows;
$$\begin{array}{ccc} V \times \cdots \times V & \longrightarrow & T^{p} \ \Psi & & \Psi \ \left(v_{1}, \ldots, v_{p}\right) & \mapsto & v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(p)^{*}} \end{array}$$

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

(a) 张量的分量及其变换定律。让 $V$ 做一个 $k$ 向量空间和让 $V$ 成为对偶空间 $V$. 在本节中，我 们将描述张量空间的属性
$$T_{q}^{p}(V)=\underbrace{V \otimes \cdots \otimes V} p \text { factors } \otimes \underbrace{V \otimes \cdots \otimes V^{*}} q \text { factors }$$

$$e_{i_{1}} \otimes \cdots \otimes e_{i_{p}} \otimes f^{j_{1}} \otimes \cdots \otimes f^{j_{q}}=t_{i_{1}, \cdots, i_{p}}^{j_{1}, \cdots, j_{q}} .$$

$$z=\sum_{i_{1}, \ldots, i_{p}, j_{1}, \ldots, j_{q}} \xi_{j_{1}, \ldots, j_{q}}^{i_{1}, \ldots, i_{p}} t_{i_{1}, \ldots, i_{p}}^{j_{1}, \ldots, j_{q}} \quad\left(\xi_{j_{1}, \ldots, j_{q}}^{i_{1}, \ldots, i_{p}} \in k\right) .$$

## 线性代数作业代写linear algebra代考|Symmetric tensors and alternating tensors

(a) 定义。有一些张量族被称为对称或交替。在本节中，我们给出它们的定义并研究它们的 性质。首先，我们考虑空间 $T^{p}(V)$ 度的逆变张量 $p$. 在本节中，我们写 $T^{p}(V)=T^{p}$ 为简单 起见。

$$P_{\sigma}\left(v_{1} \otimes \cdots \otimes v_{p}\right)=v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(p)} \quad\left(v_{i} \in V\right) .$$

(2) 对于 $\sigma, \tau \in \mathfrak{S} p, P \sigma P_{\tau}=P_{\sigma \tau}$.

$$P_{1}=I \quad\left(=\text { the identity transformation of } T^{p}\right) .$$

$$V \times \cdots \times V \quad \longrightarrow \quad T^{p} \Psi \quad \Psi\left(v_{1}, \ldots, v_{p}\right) \mapsto v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(p)^{*}}$$

# 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