# 线性代数网课代修|张量代数代写Tensor algebra辅导|MATH3349

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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 Lie algebras

(a) Definition of nilpotent Lie algebras and solvable Lie algebras. Now, let us come back to the discussion of Lie algebras and define several important families of Lie algebras. Let $\mathfrak{g}$ be a Lie algebra and $a_{1}, \mathfrak{a}_{2}$ ideals of $\mathfrak{g}$.

Denote by $\left[a_{1}, a_{2}\right]$ the subspace generated by the elements $[X, Y], X \in a_{1}$, $Y \in a_{2}$, namely,
$$\left[a_{1}, a_{2}\right]=\left{\sum_{i}\left[X_{i}, Y_{i}\right] \mid X_{i} \in a_{1}, Y_{i} \in a_{2}\right} \text {. }$$
Then $\left[a_{1}, a_{2}\right]$ is an ideal of $g$, because of the identity $[Z,[X, Y]]=$ $[[Z, X], Y]+[X,[Z, Y]]$, which is obtained from the Jacobi identity.
For an integer $m \geq 0$, we define the ideals $\mathrm{g}^{(m)}$ and $\mathfrak{g}{m}$ inductively by the following formulas: $$\mathfrak{g}^{(0)}=g{0}=\mathfrak{g}$$
$$\mathfrak{g}^{(m)}=\left[\mathfrak{g}^{(m-1)}, \mathfrak{g}^{(m-1)}\right], \quad \mathfrak{g}{m}=\left[g, \mathfrak{g}{m-1}\right] \quad(m \geq 1) .$$
It follows clearly $\mathfrak{g}{m} \supseteqq \mathfrak{g}^{(m)}$. DEFINITION 4.6. (1) If $\mathrm{g}^{(1)}=g{(1)}={0}$, then $\mathrm{g}$ is called commutative or abelian).
(2) If there is an integer $m \geq 0$ such that $\mathfrak{g}_{m}={0}$, then $\mathfrak{g}$ is called nilpotent.
(3) If there exists an integer $m \geq 0$ such that $\mathfrak{g}^{(m)}={0}$, then $\mathfrak{g}$ is called solvable.

EXAMPLE 4.11. Let $g$ be a $k$-vector space. Define a multiplication [, ]: $\mathrm{g} \times \mathrm{g} \rightarrow \mathrm{g}$ to be the zero mapping $(X, Y) \mapsto 0$, then $\mathfrak{g}$ satisfies the conditions of Lie algebras (Definition $4.1(2)$ ) and is a commutative Lie algebra.

From condition (2)(i) of Definition 4.1, Lie algebras of dimension 1 are always commutative.

## 线性代数作业代写linear algebra代考|Exercises

1. Define a Lie subalgebra $g$ of $g_{2}(k)$ by:
$$\mathfrak{g}=\left{\left[\begin{array}{ll} a & b \ 0 & c \end{array}\right] \mid a, b, c \in k\right} .$$
(1) Choose a basis for $\mathrm{g}$. Let $A$ be the matrix of a linear transformation $F$ of $\mathfrak{g}$. Give necessary and sufficient conditions with respect to $A$ for $F$ to be a derivation of $g$.
(1) Using (1), give the dimension of the Lie algebra $\operatorname{Der}(\mathfrak{g})$. Compute the dimension of the ideal ad(g) of $\operatorname{Der}(\mathfrak{g})$.
2. Let $V=\mathbf{C}^{3}$ (the complex vector space consisting of all the column vectors in $\mathbf{C}$ of dimension 3) and let $F_{i}$ be the linear transformation $\left(V \ni v \mapsto A_{i} v \in V\right)$ of $V$ defined by the matrix $A_{i}(i=1,2,3,4)$. Give a basis for the vector space of replicas of $F_{i}$.
(1) $A_{1}=\left[\begin{array}{lll}0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0\end{array}\right]$
(2) $A_{2}=\left[\begin{array}{ccc}1 & 0 & 0 \ 0 & \sqrt{2} & 0 \ 0 & 0 & 1+\sqrt{2}\end{array}\right]$,
(3) $A_{3}=\left[\begin{array}{ccc}\sqrt{2} & 1 & 0 \ 0 & \sqrt{2} & 0 \ 0 & 0 & 1\end{array}\right]$,
(4) $A_{4}=\left[\begin{array}{ccc}6 & -3 & -2 \ 4 & -1 & -2 \ 3 & -2 & 0\end{array}\right]$.
3. Let $\mathfrak{g}$ be defined as follows:
(1) Prove that $g$ is a Lie subalgebra of $g_{n}(k)$.
(2) Prove that $g$ is an algebraic Lie algebra.
(3) In general, show that the same assertions for
$$\left{\left[\begin{array}{cccc} A_{1} & & & * \ & A_{2} & & \ & & \ddots & \ & 0 & & A_{r} \end{array}\right] \in \mathfrak{g l}{n}(k) \mid \begin{array}{rr} A{i} \in \mathfrak{g l}{n{i}}(k), & i=1, \ldots, r \ & \left(n=n_{1}+\cdots+n_{r}\right) . \end{array}\right}$$

## 线性代数作业代写linear algebra代考|Properties of Lie algebras

(a) 幂零李代数和可解李代数的定义。现在，让我们回到李代数的讨论，定义几个重要的李 代数族。让 $\mathfrak{g}$ 是李代数和 $a_{1}, \mathfrak{a}{2}$ 的理想 $\mathfrak{g}$. 表示为 $\left[a{1}, a_{2}\right]$ 元素生成的子空间 $[X, Y], X \in a_{1}, Y \in a_{2}$ ，即，

$$\mathfrak{g}^{(0)}=g 0=\mathfrak{g}$$
$$\mathfrak{g}^{(m)}=\left[\mathfrak{g}^{(m-1)}, \mathfrak{g}^{(m-1)}\right], \quad \mathfrak{g} m=[g, \mathfrak{g} m-1] \quad(m \geq 1) .$$

(2) 如果有整数 $m \geq 0$ 这样 $\mathfrak{g}_{m}=0$ ，然后 $\mathfrak{g}$ 称为幂零。
(3) 如果存在整数 $m \geq 0$ 这样 $\mathfrak{g}^{(m)}=0$ ，然后 $\mathfrak{g}$ 称为可解。

## 线性代数作业代写linear algebra代考|Exercises

1. 定义李子代数 $g$ 的 $g_{2}(k)$ 经过:
2. $\backslash$ mathfrak ${g}=\backslash$ left ${\backslash$ left $[$ begin ${$ array $}{u}$ a \& $b \backslash 0$ \& $c$ lend ${$ array $} \backslash$ right $] \backslash$ mid $a, b, c \backslash$ in $k \backslash$ right $}$ 。
3. (1) 选择依据g. 让 $A$ 是一个线性变换的矩阵 $F$ 的 $\mathfrak{g}$. 给出关于的充要条件 $A$ 为了 $F$ 成为的 推导 $g$.
4. (1) 使用 (1)，给出李代数的维数Der (g). 计算理想广告的尺寸 (g) $\operatorname{Der}(\mathfrak{g})$.
5. 让 $V=\mathbf{C}^{3}$ (由所有列向量组成的复向量空间 $\mathbf{C}$ 维度 3) 并让 $F_{i}$ 是线性变换
6. $\left(V \ni v \mapsto A_{i} v \in V\right)$ 的 $V$ 由矩阵定义 $A_{i}(i=1,2,3,4)$. 给出副本的向量空间的基 $F_{i}$.
7. (1) $A_{1}=\left[\begin{array}{llllllll}0 & 1 & 0 & 0 & 0 & 11 & 0 & 0\end{array}\right]$
8. (4) $A_{4}=\left[\begin{array}{lllllll}6 & -3 & -24 & -1 & -23 & -2 & 0\end{array}\right]$.
9. 让定义如下:
10. (1) 证明 $g$ 是一个李子代数 $g_{n}(k)$.
11. (2) 证明 $g$ 是代数李代数。
12. (3) 一般来说，证明相同的断言对于

# 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