# 线性代数网课代修|高阶线性代数代写Advanced Linear Algebra代考|MATH2922

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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|Definition, examples, and notion of associated matrices

Let $U$ and $V$ be two vector spaces over the same field $\mathbb{F}$. A linear mapping or linear map or linear operator is a correspondence $T$ from $U$ into $V$, written as $T: U \rightarrow V$, satisfying the following.
(1) (Additivity) $T\left(u_{1}+u_{2}\right)=T\left(u_{1}\right)+T\left(u_{2}\right), u_{1}, u_{2} \in U$.
(2) (Homogeneity) $T(a u)=a T(u), a \in \mathbb{F}, u \in U$.

A special implication of the homogeneity condition is that $T(0)=0$. One may also say that a linear mapping ‘respects’ or preserves vector addition and scalar multiplication.

The set of all linear mappings from $U$ into $V$ will be denoted by $L(U, V)$. For $S, T \in L(U, V)$, we define $S+T$ to be a mapping from $U$ into $V$ satisfying
$$(S+T)(u)=S(u)+T(u), \quad \forall u \in U$$
For any $a \in \mathbb{F}$ and $T \in L(U, V)$, we define $a T$ to be the mapping from $U$ into $V$ satisfying
$$(a T)(u)=a T(u), \quad \forall u \in U$$
We can directly check that the mapping addition (2.1.1) and scalar-mapping multiplication (2.1.2) make $L(U, V)$ a vector space over $\mathbb{F}$. We adopt the notation $L(U)=L(U, U)$.

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

Let $U, V, W$ be vector spaces over a field $\mathbb{F}$ of respective dimensions $n, l, m$. For $T \in L(U, V)$ and $S \in L(V, W)$, we can define the composition of $T$ and $S$ with the understanding
$$(S \circ T)(x)=S(T(x)), \quad x \in U .$$
It is obvious that $S \circ T \in L(U, W)$. We now investigate the matrix of $S \circ T$ in terms of the matrices of $S$ and $T$.

To this end, let $\mathcal{B}{U}=\left{u{1}, \ldots, u_{n}\right}, \mathcal{B}{V}=\left{v{1}, \ldots, v_{l}\right}, \mathcal{B}{W}=\left{w{1}, \ldots, w_{m}\right}$ be the bases of $U, V, W$, respectively, and $A=\left(a_{i j}\right) \in \mathbb{F}(l, n)$ and $B=$ $\left(b_{i j}\right) \in \mathbb{F}(m, l)$ be the correspondingly associated matrices of $T$ and $S$, respectively. Then we have
\begin{aligned} (S \circ T)\left(u_{j}\right) &=S\left(T\left(u_{j}\right)\right)=S\left(\sum_{i=1}^{l} a_{i j} v_{i}\right)=\sum_{i=1}^{l} a_{i j} S\left(v_{i}\right) \ &=\sum_{i=1}^{l} \sum_{k=1}^{m} a_{i j} b_{k i} w_{k}=\sum_{i=1}^{m} \sum_{k=1}^{l} b_{i k} a_{k j} w_{i} \end{aligned}
In other words, we see that if we take $C=\left(c_{i j}\right) \in \mathbb{F}(m, n)$ to be the matrix associated to the linear mapping $S \circ T$ with respect to the bases $\mathcal{B}{U}$ and $\mathcal{B}{W}$, then $C=B A$. Hence the composition of linear mappings corresponds to the multiplication of their associated matrices, in the same order. For this reason, it is also customary to use $S T$ to denote $S \circ T$, when there is no risk of confusion.

## 线性代数作业代写linear algebra代考|Definition, examples, and notion of associated matrices

(1) (加法) $T\left(u_{1}+u_{2}\right)=T\left(u_{1}\right)+T\left(u_{2}\right), u_{1}, u_{2} \in U$.
(2) (同质性) $T(a u)=a T(u), a \in \mathbb{F}, u \in U$.

$$(S+T)(u)=S(u)+T(u), \quad \forall u \in U$$

$$(a T)(u)=a T(u), \quad \forall u \in U$$

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

$$(S \circ T)(x)=S(T(x)), \quad x \in U .$$

$\backslash$ Imathcal ${B}{U}=\backslash$ left $\left{u{1}, \backslash\right.$ dots, $u_{-}{n} \backslash$ right $}$, \mathcal ${B}{V}=\backslash$ left $\left{v{1}, \backslash\right.$ dots, $v_{-}{l} \backslash$ right $}, \backslash$ mathcal ${B}{W}=\backslash l e f t{w{1}, \backslash l d$

$$(S \circ T)\left(u_{j}\right)=S\left(T\left(u_{j}\right)\right)=S\left(\sum_{i=1}^{l} a_{i j} v_{i}\right)=\sum_{i=1}^{l} a_{i j} S\left(v_{i}\right) \quad=\sum_{i=1}^{l} \sum_{k=1}^{m} a_{i j} b_{k i} w_{k}=\sum_{i=1}^{m} \sum_{k=1}^{l} b_{i k} a_{k j} w_{i}$$

# 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