# 线性代数作业代写linear algebra代考|Introduction

my-assignmentexpert™ 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|Subspaces of F^

A subset $S$ of $F^{n}$ is called a subspace of $F^{n}$ if

1. The zero vector belongs to $S$; (that is, $0 \in S$ );
2. If $u \in S$ and $v \in S$, then $u+v \in S$; (S is said to be closed under vector addition);
3. If $u \in S$ and $t \in F$, then $t u \in S ;(S$ is said to be closed under scalar multiplication).

Let $A \in M_{m \times n}(F)$. Then the set of vectors $X \in F^{n}$ satisfying $A X=0$ is a subspace of $F^{n}$ called the null space of $A$ and is denoted here by $N(A)$. (It is sometimes called the solution space of $A$.)
Proof. (1) $A 0=0$, so $0 \in N(A)$; (2) If $X, Y \in N(A)$, then $A X=0$ and $A Y=0$, so $A(X+Y)=A X+A Y=0+0=0$ and so $X+Y \in N(A)$; (3) If $X \in N(A)$ and $t \in F$, then $A(t X)=t(A X)=t 0=0$, so $t X \in N(A)$.
For example, if $A=\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right]$, then $N(A)={0}$, the set consisting of just the zero vector. If $A=\left[\begin{array}{ll}1 & 2 \ 2 & 4\end{array}\right]$, then $N(A)$ is the set of all scalar multiples of $[-2,1]^{t}$.

## 线性代数作业代写linear algebra代考|consisting of all linear combinations

Let $X_{1}, \ldots, X_{m} \in F^{n}$. Then the set consisting of all linear combinations $x_{1} X_{1}+\cdots+x_{m} X_{m}$, where $x_{1}, \ldots, x_{m} \in F$, is a subspace of $F^{n}$. This subspace is called the subspace spanned or generated by $X_{1}, \ldots, X_{m}$ and is denoted here by $\left\langle X_{1}, \ldots, X_{m}\right\rangle$. We also call $X_{1}, \ldots, X_{m}$ a spanning family for $S=\left\langle X_{1}, \ldots, X_{m}\right\rangle$.

Proof. (1) $0=0 X_{1}+\cdots+0 X_{m}$, so $0 \in\left\langle X_{1}, \ldots, X_{m}\right\rangle$; (2) If $X, Y \in$ $\left\langle X_{1}, \ldots, X_{m}\right\rangle$, then $X=x_{1} X_{1}+\cdots+x_{m} X_{m}$ and $Y=y_{1} X_{1}+\cdots+y_{m} X_{m}$, so
\begin{aligned} X+Y &=\left(x_{1} X_{1}+\cdots+x_{m} X_{m}\right)+\left(y_{1} X_{1}+\cdots+y_{m} X_{m}\right) \ &=\left(x_{1}+y_{1}\right) X_{1}+\cdots+\left(x_{m}+y_{m}\right) X_{m} \in\left\langle X_{1}, \ldots, X_{m}\right\rangle \end{aligned}
(3) If $X \in\left\langle X_{1}, \ldots, X_{m}\right\rangle$ and $t \in F$, then
\begin{aligned} X &=x_{1} X_{1}+\cdots+x_{m} X_{m} \ t X &=t\left(x_{1} X_{1}+\cdots+x_{m} X_{m}\right) \ &=\left(t x_{1}\right) X_{1}+\cdots+\left(t x_{m}\right) X_{m} \in\left\langle X_{1}, \ldots, X_{m}\right\rangle \end{aligned}
For example, if $A \in M_{m \times n}(F)$, the subspace generated by the columns of $A$ is an important subspace of $F^{m}$ and is called the column space of $A$. The column space of $A$ is denoted here by $C(A)$. Also the subspace generated by the rows of $A$ is a subspace of $F^{n}$ and is called the row space of $A$ and is denoted by $R(A)$.

## 线性代数作业代写linear algebra代考|Subspaces of F^

1. 零向量属于小号; （那是，0∈小号);
2. 如果你∈小号和v∈小号， 然后你+v∈小号; （据说 S 在向量加法下是闭合的）；
3. 如果你∈小号和吨∈F， 然后吨你∈小号;(小号据说在标量乘法下是封闭的）。

## 线性代数作业代写linear algebra代考|consisting of all linear combinations

X+和=(X1X1+⋯+X米X米)+(和1X1+⋯+和米X米) =(X1+和1)X1+⋯+(X米+和米)X米∈⟨X1,…,X米⟩
(3) 如果X∈⟨X1,…,X米⟩和吨∈F， 然后
X=X1X1+⋯+X米X米 吨X=吨(X1X1+⋯+X米X米) =(吨X1)X1+⋯+(吨X米)X米∈⟨X1,…,X米⟩

# 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