线性代数网课代修|张量代数代写Tensor algebra辅导|MATH306

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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|Tensor products of R-modules

Modules over a ring are a generalization of the concept of vector spaces. The definition of tensor product generalizes and we present a theorem generalizing Theorem 1.1. In this section we shall discuss this generalization. These results can be easily obtained as a variation of the construction explained in $\S$ 9. We do not go into the details, since the further properties of a tensor product of modules generally depend on the properties of rings. First we give several definitions.

DEFinition 1.7. Let $P$ be a set. $P$ is called a module if there is a rule which associates with every (ordered) pair $m_{1}, m_{2}$ of elements of $P$ a third element $m_{1}+m_{2}$ of $P$, which is called the sum of $m_{1}$ and $m_{2}$, and if the following conditions are satisfied:

For all $m_{1}, m_{2}, m_{3} \in P$, $$\left(m_{1}+m_{2}\right)+m_{3}=m_{1}+\left(m_{2}+m_{3}\right) \text { (associativity). }$$ There exists an element $0 \in P$ such that, for all $m \in P$, $$m+0=0+m=m .$$ For each $m \in P$, there exists an element $m^{\prime} \in P$ such that $m+m^{\prime}=$ $m^{\prime}+m=0$. ( $m^{\prime}$ is denoted by $-m$.) For all $m_{1}, m_{2} \in P$, $$m_{1}+m_{2}=m_{2}+m_{1} \quad \text { (commutativity). }$$ DEfINITION 1.8. Let $R$ be a set. $R$ is called a ring if there are two rules which associate with each ordered pair $r_{1}, r_{2}$ of $R$, their sum $r_{1}+r_{2} \in R$ and product $r_{1} r_{2} \in R$ and if the following conditions are satisfied: $R$ is a module with respect to addition.

线性代数作业代写linear algebra代考|Definition and examples of tensors

Let $V$ be a $k$-vector space and let $V^{}$ be the dual space of $V$. Let $T$ be a vector space which is obtained as the tensor product of several copies of $V$ and of $V^{}$, for example $(V \otimes V) \otimes V^{}$ or $V \otimes V^{} \otimes V^{*} \otimes V$. Then $T$ is generally called a tensor space.

From the associativity of the tensor product (Proposition 1.8) we can, when we make a tensor space, neglect parentheses. Therefore a tensor space is isomorphic to some $V^{\varepsilon_{1}} \otimes \cdots \otimes V^{\varepsilon_{r}}$, the tensor product of a sequence $V^{\varepsilon_{1}}, \ldots, V^{\varepsilon_{r}}$ consisting of $V$ and $V^{}$, where we set $V=V^{1}, V^{}=V^{-1}$, and $\varepsilon_{i}=\pm 1$. Elements of this space are called tensors of type $\left(\varepsilon_{1}, \ldots, \varepsilon_{r}\right)$. We can make a further identification. Suppose that $T$ is isomorphic to $V^{\varepsilon_{1}} \otimes \cdots \otimes V^{\varepsilon_{r}}$ and $T^{\prime}$ is isomorphic to $V^{\varepsilon_{1}^{\prime}} \otimes \cdots \otimes V^{\varepsilon_{r}^{\prime}}$. (Both have the same number of factors.) If the number of $V$ among $V^{\varepsilon_{1}}, \ldots, V^{\varepsilon_{r}}$ and that among $V^{\varepsilon_{1}^{\prime}}, \ldots, V^{\varepsilon_{r}^{\prime}}$ are equal (so the numbers of $V^{}$ are also equal), then there is a natural isomorphism between $T$ and $T^{\prime}$ obtained from the commutativity of the tensor product (Proposition $1.5, V \otimes V^{} \cong V^{} \otimes V$ ), and we can identify $T$ and $T^{\prime}$ (for example, $T=V \otimes V \otimes V^{} \otimes V$ and $\left.T^{\prime}=V^{} \otimes V \otimes V \otimes V\right)$. Therefore, to discuss a tensor space $T$, the numbers of $V$ and $V^{}$ factors are important. If $T$ is a tensor product of $p$ copies of $V$ and $q$ copies of $V^{}$, we call $T$ the tensor space of type $(p, q)$ and denote it by $T_{q}^{p}(V)$. Thus, if we identify as above, $$T_{q}^{p}(V)=\underbrace{V \otimes \cdots \otimes V}{p \text { factors }} \otimes \underbrace{V^{} \otimes \cdots \otimes V^{*}}{q \text { factors }}$$

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