linearalgebra.me 为您的留学生涯保驾护航 在线性代数linear algebra作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的线性代数linear algebra代写服务。我们的专家在线性代数linear algebra代写方面经验极为丰富，各种线性代数linear algebra相关的作业也就用不着 说。

• 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|Nilpotent mappings

Consider the vector space $\mathcal{P}{n}$ of the set of all polynomials of degrees up to $n$ with coefficients in a field $\mathbb{F}$ and the differentiation operator $$D=\frac{\mathrm{d}}{\mathrm{d} t} \text { sô that } D\left(a{0}+a_{1} l+\cdots+a_{n} l^{n}\right)=a_{1}+2 a_{2} l+\cdots+n a_{n} l^{n-1} \text {. }$$ Then $D^{n+1}=0$ (zero mapping). Such a linear mapping $D: \mathcal{P}{n} \rightarrow \mathcal{P}{n}$ is an example of nilpotent mappings we now study.

Definition $2.18$ Let $U$ be a finite-dimensional vector space and $T \in L(U)$. We say that $T$ is nilpotent if there is an integer $k \geq 1$ such that $T^{k}=0$. For a nilpotent mapping $T \in L(U)$, the smallest integer $k \geq 1$ such that $T^{k}=0$ is called the degree or index of nilpotence of $T$. The same definition may be stated for square matrices. Of course, the degree of a nonzero nilpotent mapping is always at least $2 .$ Definition 2.19 Let $U$ be a vector space and $T \in L(U)$. For any nonzero vector $u \in U$, we say that $u$ is $T$-cyclic if there is an integer $m \geq 1$ such that $T^{m}(u)=0$. The smallest such integer $m$ is called the period of $u$ under or relative to $T$. If each vector in $U$ is $T$-cyclic, $T$ is said to be locally nilpotent. It is clear that a nilpotent mapping must be locally nilpotent. In fact, these two notions are equivalent in finite dimensions.

## 线性代数作业代写linear algebra代考|Polynomials of linear mappings

In this section, we have seen that it is often useful to consider various powers of a linear mapping $T \in L(U)$ as well as some linear combinations of appropriate powers of $T$. These manipulations motivate the introduction of the notion of polynomials of linear mappings. Specifically, for any $p(t) \in \mathcal{P}$ with the form $$p(t)=a_{n} t^{n}+\cdots+a_{1} t+a_{0}, \quad a_{0}, a_{1}, \ldots, a_{n} \in \mathbb{F},$$ we define $p(T) \in L(U)$ to be the linear mapping over $U$ given by $$p(T)=a_{n} T^{n}+\cdots+a_{1} T+a_{0} I$$ It is straightforward to check that all usual operations over polynomials in variable $t$ can be carried over correspondingly to those over polynomials in the powers of a linear mapping $T$ over the vector space $U$. For example, if $f, g, h \in \mathcal{P}$ satisfy the relation $f(t)=g(t) h(t)$, then $$f(T)=g(T) h(T)$$ because the powers of $T$ follow the same rule as the powers of $t$. That is, $T^{k} T^{l}=T^{k+l}, k, l \in \mathbb{N}$.

For $T \in L(U)$, let $\lambda \in \mathbb{F}$ be an eigenvalue of $T$. Then, for any $p(t) \in \mathcal{P}$ given as in (2.5.48), $p(\lambda)$ is an eigenvalue of $p(T)$. To see this, we assume that $u \in U$ is an eigenvector of $T$ associated with $\lambda$. We have \begin{aligned} p(T)(u) &=\left(a_{n} T^{n}+\cdots+a_{1} T+a_{0} I\right)(u) \ &=\left(a_{n} \lambda^{n}+\cdots+a_{1} \lambda+a_{0}\right)(u)=p(\lambda) u \end{aligned} as anticipated.

## 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