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

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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|independence

We now recall the definition of linear dependence and independence of a family of vectors in $F^{n}$ given in Chapter $2 .$

DEFINITION 3.3.1 Vectors $X_{1}, \ldots, X_{m}$ in $F^{n}$ are said to be linearly dependent if there exist scalars $x_{1}, \ldots, x_{m}$, not all zero, such that
$$x_{1} X_{1}+\cdots+x_{m} X_{m}=0$$
In other words, $X_{1}, \ldots, X_{m}$ are linearly dependent if some $X_{i}$ is expressible as a linear combination of the remaining vectors.
$X_{1}, \ldots, X_{m}$ are called linearly independent if they are not linearly dependent. Hence $X_{1}, \ldots, X_{m}$ are linearly independent if and only if the equation
$$x_{1} X_{1}+\cdots+x_{m} X_{m}=0$$
has only the trivial solution $x_{1}=0, \ldots, x_{m}=0$.
EXAMPLE 3.3.1 The following three vectors in $\mathbb{R}^{3}$
$$X_{1}=\left[\begin{array}{l} 1 \ 2 \ 3 \end{array}\right], \quad X_{2}=\left[\begin{array}{r} -1 \ 1 \ 2 \end{array}\right], \quad X_{3}=\left[\begin{array}{r} -1 \ 7 \ 12 \end{array}\right]$$
are linearly dependent, as $2 X_{1}+3 X_{2}+(-1) X_{3}=0$.

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

REMARK 3.3.1 If $X_{1}, \ldots, X_{m}$ are linearly independent and
$$x_{1} X_{1}+\cdots+x_{m} X_{m}=y_{1} X_{1}+\cdots+y_{m} X_{m},$$
then $x_{1}=y_{1}, \ldots, x_{m}=y_{m}$. For the equation can be rewritten as
$$\left(x_{1}-y_{1}\right) X_{1}+\cdots+\left(x_{m}-y_{m}\right) X_{m}=0$$
and so $x_{1}-y_{1}=0, \ldots, x_{m}-y_{m}=0$.
THEOREM 3.3.1 A family of $m$ vectors in $F^{n}$ will be linearly dependent if $m>n$. Equivalently, any linearly independent family of $m$ vectors in $F^{n}$ must satisfy $m \leq n$.
Proof. The equation
$$x_{1} X_{1}+\cdots+x_{m} X_{m}=0$$
is equivalent to $n$ homogeneous equations in $m$ unknowns. By Theorem 1.5.1, such a system has a non-trivial solution if $m>n$.

The following theorem is an important generalization of the last result and is left as an exercise for the interested student:

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

X1X1+⋯+X米X米=0

X1,…,X米如果它们不是线性相关的，则称为线性无关。因此X1,…,X米当且仅当方程是线性独立的
X1X1+⋯+X米X米=0

X1=[1 2 3],X2=[−1 1 2],X3=[−1 7 12]

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

X1X1+⋯+X米X米=和1X1+⋯+和米X米,

(X1−和1)X1+⋯+(X米−和米)X米=0

X1X1+⋯+X米X米=0

# 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