# 线性代数作业代写linear algebra代考|Rank and nullity of a matrix

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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|Every linearly independent

Every linearly independent family of vectors in a subspace $S$ can be extended to a basis of $S$.

Proof. Suppose $S$ has basis $X_{1}, \ldots, X_{m}$ and that $Y_{1}, \ldots, Y_{r}$ is a linearly independent family of vectors in $S$. Then
$$S=\left\langle X_{1}, \ldots, X_{m}\right\rangle=\left\langle Y_{1}, \ldots, Y_{r}, X_{1}, \ldots, X_{m}\right\rangle$$
as each of $Y_{1}, \ldots, Y_{r}$ is a linear combination of $X_{1}, \ldots, X_{m}$.
Then applying the left-to-right algorithm to the second spanning family for $S$ will yield a basis for $S$ which includes $Y_{1}, \ldots, Y_{r}$.

## 线性代数作业代写linear algebra代考|Given that the reduced row–echelon form of

Given that the reduced row-echelon form of
$$A=\left[\begin{array}{rrrrr} 1 & 1 & 5 & 1 & 4 \ 2 & -1 & 1 & 2 & 2 \ 3 & 0 & 6 & 0 & -3 \end{array}\right]$$
equal to
$$B=\left[\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -1 \ 0 & 1 & 3 & 0 & 2 \ 0 & 0 & 0 & 1 & 3 \end{array}\right]$$
find bases for $R(A), C(A)$ and $N(A)$.
Solution. $[1,0,2,0,-1],[0,1,3,0,2]$ and $[0,0,0,1,3]$ form a basis for $R(A)$. Also
$$A_{* 1}=\left[\begin{array}{l} 1 \ 2 \ 3 \end{array}\right], A_{* 2}=\left[\begin{array}{r} 1 \ -1 \ 0 \end{array}\right], A_{* 4}=\left[\begin{array}{l} 1 \ 2 \ 0 \end{array}\right]$$
form a basis for $C(A)$.
Finally $N(A)$ is given by
$$\left[\begin{array}{l} x_{1} \ x_{2} \ x_{3} \ x_{4} \ x_{5} \end{array}\right]=\left[\begin{array}{c} -2 x_{3}+x_{5} \ -3 x_{3}-2 x_{5} \ x_{3} \ -3 x_{5} \ x_{5} \end{array}\right]=x_{3}\left[\begin{array}{r} -2 \ -3 \ 1 \ 0 \ 0 \end{array}\right]+x_{5}\left[\begin{array}{r} 1 \ -2 \ 0 \ -3 \ 1 \end{array}\right]=x_{3} X_{1}+x_{5}$$
where $x_{3}$ and $x_{5}$ are arbitrary. Hence $X_{1}$ and $X_{2}$ form a basis for $N(A)$. Here $\operatorname{rank} A=3$ and nullity $A=2$.

## 线性代数作业代写linear algebra代考|Given that the reduced row–echelon form of

[X1 X2 X3 X4 X5]=[−2X3+X5 −3X3−2X5 X3 −3X5 X5]=X3[−2 −3 1 0 0]+X5[1 −2 0 −3 1]=X3X1+X5

# 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