# 线性代数作业代写linear algebra代考|Basis of a subspace

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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|belonging

We now come to the important concept of basis of a vector subspace.
DEFINITION 3.4.1 Vectors $X_{1}, \ldots, X_{m}$ belonging to a subspace $S$ are said to form a basis of $S$ if
(a) Every vector in $S$ is a linear combination of $X_{1}, \ldots, X_{m}$;
(b) $X_{1}, \ldots, X_{m}$ are linearly independent.
Note that (a) is equivalent to the statement that $S=\left\langle X_{1}, \ldots, X_{m}\right\rangle$ as we automatically have $\left\langle X_{1}, \ldots, X_{m}\right\rangle \subseteq S$. Also, in view of Remark 3.3.1 above, (a) and (b) are equivalent to the statement that every vector in $S$ is uniquely expressible as a linear combination of $X_{1}, \ldots, X_{m}$.

## 线性代数作业代写linear algebra代考|subspace

THEOREM 3.4.2 Any two bases for a subspace $S$ must contain the same number of elements.

Proof. For if $X_{1}, \ldots, X_{r}$ and $Y_{1}, \ldots, Y_{s}$ are bases for $S$, then $Y_{1}, \ldots, Y_{s}$ form a linearly independent family in $S=\left\langle X_{1}, \ldots, X_{r}\right\rangle$ and hence $s \leq r$ by Theorem 3.3.2. Similarly $r \leq s$ and hence $r=s$.

DEFINITION 3.4.2 This number is called the dimension of $S$ and is written $\operatorname{dim} S$. Naturally we define $\operatorname{dim}{0}=0$.

It follows from Theorem 3.3.1 that for any subspace $S$ of $F^{n}$, we must have $\operatorname{dim} S \leq n .$

EXAMPLE 3.4.3 If $E_{1}, \ldots, E_{n}$ denote the $n$-dimensional unit vectors in $F^{n}$, then $\operatorname{dim}\left\langle E_{1}, \ldots, E_{i}\right\rangle=i$ for $1 \leq i \leq n$.
The following result gives a useful way of exhibiting a basis.

## 线性代数作业代写linear algebra代考|belonging

(a) 中的每个向量小号是一个线性组合X1,…,X米;
(二)X1,…,X米是线性独立的。

# 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