# 线性代数网课代修|张量代数代写Tensor algebra辅导|MA26500

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• 数值分析
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• 矩阵论
• 优化理论
• 线性规划
• 逼近论

Let $A$ be a $k$-vector space and $m$ a bilinear mapping $m$ of $A \times A$ into $A$. As we explained in $\$ 2.1$Example 2.4, we can define a multiplication on$A$by calling$m\left(a_{1}, a_{2}\right)$the product of$a_{1}$and$a_{2}$. In this case$A$(more exactly, the pair$(A, m))$is called an algebra over$k$. In the following, we sometimes write$m\left(a_{1}, a_{2}\right)=a_{1} \cdot a_{2}$for simplicity. An associative algebra-in$\S 2.4(\mathrm{~b})$– and a Lie algebra whose definition we give below, are interesting examples of algebras. For an algebra in general, using the mapping$m$, we can define homomorphism, isomorphism, subalgebra, ideal, factor algebra, and other properties in a similar fashion as in the case of an associative algebra. Let$A$be an algebra of dimension$n$and$\mathscr{E}=\left(e_{1}, \ldots, e_{n}\right)$a basis of$A$. The product$m\left(e_{i}, e_{j}\right)$can be expressed as a linear combination of basis elements, $$m\left(e_{i}, e_{j}\right)=\sum_{r=1}^{n} \alpha_{i j}{ }^{r} e_{r} \quad\left(\alpha_{i j}{ }^{r} \in k\right) .$$ The$n^{3}$elements$\alpha_{i j}{ }^{r}$of$k$are called the structure constants of$A$with respect to the basis$\mathscr{E}$. We shall show below that they are the components of$m$with respect to$\mathscr{E}$, when$m$is regarded as tensor of type$(1,2)$. By the isomorphism$\mathscr{L}(A, A ; A) \cong T_{2}^{1}(A)$explained in Example 2.4,$\S 2.1$, to a standard basis element$t_{r}^{p q} \quad\left(=f^{p} \otimes f^{q} \otimes e_{r}\right.$, where$\left(f^{1}, \ldots, f^{n}\right)$is the dual basis of$\mathscr{E}$), there corresponds a bilinear mapping of$A \times A$into$A$such that$\left(e_{i}, e_{j}\right) \mapsto f^{p}\left(e_{i}\right) f^{q}\left(e_{j}\right) e_{r}=\delta_{p i} \delta_{q j} e_{r}$(where$\delta_{i j}$denotes Kronecker’s symbol). Thus, if we express the mapping$m$by standard basis elements as $$m=\sum_{p, q, r} \xi_{p q}{ }^{r} t_{r}^{p q} \quad\left(\xi_{p q}{ }^{r} \in k\right),$$ we have $$m\left(e_{i}, e_{j}\right)=\sum_{p, q, r} \xi_{p q}{ }^{r} \delta_{p i} \delta_{q j} e_{r}=\sum_{r} \xi_{i j}{ }^{r} e_{r},$$ which shows$\alpha_{i j}{ }^{r}=\xi_{i j}{ }^{r}$. ## 线性代数作业代写linear algebra代考|Replicas of matrices In order to prepare for the further discussion of Lie algebras, we introduce the theory$\left({ }^{1}\right)$of replicas of a matrix originated by Chevalley. We start with the following results on matrices. As before$k$denotes a field of characteristic$0 .$THEOREM I. Let$A$be an$n \times n$matrix with coefficients in$k . \quad Z(A)$denotes the set of matrices which commute with$A$. Then a matrix$X$which commutes with all elements of$Z(A)$can be expressed as a polynomial in A. Namely, there exists a polynomial$P(x)=\sum_{i=0}^{n} \alpha_{i} x^{i} \quad\left(\alpha_{i} \in k\right)$such that$X=\sum_{i=0}^{n} \alpha_{i} A^{i}$. For a proof, see, e.g., N. Jacobson, Lectures in abstract algebra, vol. II, van Nostrand (republished by Springer), p.$113 .$Definition 4.3. An$n \times n$matrix$A$is said to be semisimple if it is similar to a diagonal matrix over some extension field of$k$, and nilpotent if there exists an integer$\nu$such that$A^{\nu}=0$. THEOREM II (Jordan decomposition). An$n \times n$matrix$A$can be written uniquely as a sum$A=S+N$, where$S$is semisimple,$N$is nilpotent, and$S N=N S$. Moreover,$S$and$N$can be expressed as polynomials in$A$without constant term. (That is, there are polynomials$P_{1}(x)$and$P_{2}(x)$such that$S=P_{1}(A), N=P_{2}(A)$, and$P_{1}(0)=P_{2}(0)=0$.)$S$and$N$are called the semisimple part and the nilpotent part of$A$respectively. For a proof, see, e.g., I. Satake, Linear algebra, Marcel Dekker, p. 168.$\left(^{2}\right)$Let$V$be an$n$-dimensional$k$-vector space. We can define semisimple and nilpotent linear transformations. And, for a linear transformation$A$, the semisimple part and nilpotent part of$A$are also defined and denoted by$A^{(\mathrm{s})}$and$A^{(\mathrm{n})}$respectively. They are the linear transformations whose matrices are the semisimple part and nilpotent part of the matrix of$A$with respect to any basis. ## 线性代数作业代写linear algebra代考|Algebraic systems with bilinear multiplication 让$A$做一个$k$-向量空间和$m$双线性映射$m$的$A \times A$进入$A$. 正如我们在$\$2.1$ 例 2.4，我们可以 定义一个乘法 $A$ 通过调用 $m\left(a_{1}, a_{2}\right)$ 的产物 $a_{1}$ 和 $a_{2}$. 在这种情况下 $A$ (更准确地说，这对 $(A, m))$ 被称为代数 $k$. 下面，我们有时会写 $m\left(a_{1}, a_{2}\right)=a_{1} \cdot a_{2}$ 为简单起见。

$$m\left(e_{i}, e_{j}\right)=\sum_{r=1}^{n} \alpha_{i j}{ }^{r} e_{r} \quad\left(\alpha_{i j}{ }^{r} \in k\right) .$$

$$m=\sum_{p, q, r} \xi_{p q}{ }^{r} t_{r}^{p q} \quad\left(\xi_{p q}{ }^{r} \in k\right),$$

$$m\left(e_{i}, e_{j}\right)=\sum_{p, q, r} \xi_{p q}{ }^{r} \delta_{p i} \delta_{q j} e_{r}=\sum_{r} \xi_{i j}^{r} e_{r}$$

# 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