Course: 8.5 行操作和列梯层表格

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行操作和行梯表

Applying row operations to reduce a matrix is a procedural skill that takes lots of writing, rewriting and careful arithmetic. The payoff for being able to transform a matrix into a simplified form will become clear later. For now, what does the simplified form mean for a matrix?
::应用行操作减少矩阵是一个程序技能,需要大量书写、重写和仔细算术。能够将矩阵转换为简化格式的回报将在稍后显现出来。目前,简化格式对矩阵意味着什么?

Row Operations and Row Echelon Forms
::行操作和行梯表

There are only three operations that are permitted to act on matrices. They are the exact same operations that are permitted when solving a system of equations .
::只有三种操作允许根据矩阵采取行动,它们是在解决方程系统时允许的完全相同的操作。

Add a multiple of one row to another row.
::将多个行加到另一行。

Scale a row by multiplying through by a non-zero constant .
::通过乘以非零常数来缩放一行。

Swap two rows.
::交换两排

Using these three operations, your job is to simplify matrices into row echelon form . Row echelon form must meet three requirements.
::使用这三种操作,您的工作是将矩阵简化为排梯表格。列梯表格必须满足三项要求。

1. The leading coefficient of each row must be a one.
::1. 每一行的主要系数必须是一行。

2. All entries in a column below a leading one must be zero.
::2. 前列一栏下的所有条目必须为零。

3. All rows that just contain zeros are at the bottom of the matrix.
::3. 所有只包含零的行都在矩阵的底部。

Here are some examples of matrices in row echelon form:
::以下是以行梯层形式列出的一些矩阵实例:

$\begin{align*}\begin{bmatrix} 1 & 14\\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 4 \end{bmatrix}, \begin{bmatrix} 1 & 2 & 3 & 5 & 6\\ 0 & 0 & 1 & 4 & 7\\ 0 & 0 & 0 & 1 & -2\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\end{align*}$

Reduced row echelon form also has one extra stipulation compared with row echelon form.
::减少的排梯表与排梯表相比,还有一项额外规定。

4. Every leading coefficient of 1 must be the only non-zero element in that column.
::4. 每一主要系数1必须是该栏中唯一的非零要素。

Here are some examples of matrices in reduced row echelon form :
::以下是一些缩排梯层表格的矩阵实例:

$\begin{align*}\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 & 3\\ 0 & 1 & 4 \end{bmatrix}, \begin{bmatrix} 1 & 2 & 0 & 0 & 6\\ 0 & 0 & 1 & 0 & 7\\ 0 & 0 & 0 & 1 & -2\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\end{align*}$

Putting a matrix into reduced row echelon form is a result of performing Gauss-Jordan elimination . The process illustrated in this concept is named after those two mathematicians.
::将矩阵制成减排梯式是消除高斯-约旦的结果,这个概念所说明的过程是以这两个数学家命名的。

To put the a matrix into reduced row echelon form, use the row operations to change the matrix. Take the following matrix:
::要将矩阵制成减排梯层表,请使用行操作来更改矩阵。

$\begin{align*}\begin{bmatrix} 3 & 7\\ 2 & 5 \end{bmatrix}\end{align*}$

In each step of reducing the matrix, only one of the three row operations will be used. Specific shorthand will be introduced.
::在减少矩阵的每一个步骤中,将只使用三行操作中的一个,将采用具体的简称。

\begin{aligned} [\begin{matrix} 3 & 7 \\ 2 & 5 \end{matrix}] \\ 3 R_{2} \to [\begin{matrix} 3 & 7 \\ 6 & 15 \end{matrix}] \\ - 2 R_{1} + R_{2} \to [\begin{matrix} 3 & 7 \\ 0 & 1 \end{matrix}] \\ - 7 R_{2} + R_{1} \to [\begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}] \\ \frac{1}{3} R \to [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \end{aligned}

$\begin{align*}\begin{bmatrix} 3 & 7\\ 2 & 5 \end{bmatrix}\\ 3{R}_{2} \rightarrow \begin{bmatrix} 3&7\\ 6 & 15 \end{bmatrix}\\ -2{R}_{1}+{R}_{2} \rightarrow \begin{bmatrix} 3 & 7 \\ 0 & 1 \end{bmatrix}\\ -7{R}_{2}+{R}_{1} \rightarrow \begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}\\ \frac{1}{3}R \rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\end{align*}$
::[37615]-2R1+R2}[3701]-7R2+R1}[3001]13R}[1001]

Note that the $\begin{align*}3{R}_{2}\end{align*}$ indicates that the second row of the matrix is scaled by a factor of 3. The $\begin{align*}-2{R}_{1}+{R}_{2}\end{align*}$ before the third matrix indicates that the second row has two times the first row subtracted from it.
::请注意,3R2表示,矩阵第二行的缩放因数为3。在第三个矩阵之前的-2R1+R2表示,第二行是从第二行中减去的第一行的两倍。

Row reducing a $\begin{align*}2 \times 2\end{align*}$ matrix to become the identity matrix illustrates the fact that the rows of the original matrix are linearly independent .
::将一个 2x2 矩阵缩小为身份矩阵的行说明,原始矩阵的行线性独立。

Examples
::实例

Example 1
::例1

Earlier, you were asked what it means for a matrix to be simplified. There are two forms of a matrix that are most simplified. The most important is reduced row echelon form that follows the four stipulations from the guidance section. An example of a matrix in reduced row echelon form is:
::早些时候,有人问您简化矩阵意味着什么。有两种形式最简化的矩阵。最重要的是按照指导部分的四项规定, 缩排梯层表。缩排梯层表的示例是:

\begin{aligned} [\begin{matrix} 1 & 0 & 0 & 2 & 43 \\ 0 & 1 & 0 & 2 & 3 \\ 0 & 0 & 1 & 98 & 5 \end{matrix}] \end{aligned}

$\begin{align*}\begin{bmatrix} 1 & 0 & 0 & 2 & 43\\ 0 & 1 & 0 & 2 & 3\\ 0 & 0 & 1 & 98 & 5 \end{bmatrix}\end{align*}$

Example 2
::例2

Put the following matrix into reduced row echelon form.
::将下列矩阵划入缩排的梯层表。

$\begin{align*}\begin{bmatrix} 2 & 4 & 0\\ 0 & 3 & 1\\ 1 & 2 & 4 \end{bmatrix}\end{align*}$

\begin{aligned} [\begin{matrix} 2 & 4 & 0 \\ 0 & 3 & 1 \\ 1 & 2 & 4 \end{matrix}] \\ R_{1} \cdot - \frac{1}{2} + R_{3} \to [\begin{matrix} 2 & 4 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 4 \end{matrix}] \\ R_{3} \div 4 \to [\begin{matrix} 2 & 4 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{matrix}] \\ R_{1} \div 2, R_{2} \div 3 \to [\begin{matrix} 1 & 2 & 0 \\ 0 & 1 & \frac{1}{3} \\ 0 & 0 & 1 \end{matrix}] \\ R_{3} \cdot - \frac{1}{3} + R_{2} \to [\begin{matrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \\ R_{2} \cdot - 2 + R_{1} \to [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \end{aligned}

$\begin{align*}\begin{bmatrix} 2 & 4 & 0\\ 0 & 3 & 1\\ 1 & 2 & 4 \end{bmatrix}\\ {R}_{1}\cdot -\frac{1}{2} + {R}_{3} \rightarrow \begin{bmatrix} 2 & 4 & 0\\ 0 & 3 & 1\\ 0 & 0 & 4 \end{bmatrix}\\ {R}_{3} \div 4 \rightarrow \begin{bmatrix} 2 & 4 & 0\\ 0 & 3 & 1\\ 0 & 0 & 1 \end{bmatrix}\\ {R}_{1} \div 2, \ {R}_{2} \div 3 \rightarrow \begin{bmatrix} 1 & 2 & 0\\ 0 & 1 & \frac{1}{3}\\ 0 & 0 & 1 \end {bmatrix}\\ {R}_{3} \cdot -\frac{1}{3} + {R}_{2} \rightarrow \begin{bmatrix} 1 & 2 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\\ {R}_{2} \cdot -2 + {R}_{1} \rightarrow \begin{bmatrix} 1 & 0 & 0\\ 0& 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\end{align*}$
::[240031124]R112+R3}[240031004]R34}[240031001]R34}[240031001]R12,R23}[1200111000]R313+R2}[120010001]R22+R1}[1001001001]

Note that two operations were used in the fourth row to produce the fourth matrix. This is acceptable when the operations do not interfere or interact with each other.
::请注意,第四行使用两个操作来生成第四个矩阵,当操作不相互干扰或互动时可以接受。

Again, row reducing a $\begin{align*}3 \times 3\end{align*}$ matrix to become the identity matrix is just an exercise that illustrates the fact that the rows were linearly independent.
::同样,将3x3矩阵减为身份矩阵的行数只是表明各行线性独立这一事实的一种做法。

Example 3
::例3

Reduce the following matrix to reduced row echelon form.
::将下列矩阵减为减排梯层表。

$\begin{align*}\begin{bmatrix} 0 & 4 & 5\\ 2 & 6 & 8 \end{bmatrix}\end{align*}$

\begin{aligned} [\begin{matrix} 0 & 4 & 5 \\ 2 & 6 & 8 \end{matrix}] \\ Switch Rows \to [\begin{matrix} 2 & 6 & 8 \\ 0 & 4 & 5 \end{matrix}] \\ R_{1} \div 2 \to [\begin{matrix} 1 & 3 & 4 \\ 0 & 4 & 5 \end{matrix}] \\ R_{2} \cdot \frac{1}{4} \to [\begin{matrix} 1 & 3 & 4 \\ 0 & 1 & \frac{5}{4} \end{matrix}] \\ R_{2} \cdot - 3 + R_{1} \to [\begin{matrix} 1 & 0 & \frac{1}{4} \\ 0 & 1 & \frac{5}{4} \end{matrix}] \end{aligned}

$\begin{align*}\begin{bmatrix} 0 & 4 & 5\\ 2 & 6 & 8 \end{bmatrix}\\ \text {Switch Rows} \rightarrow \begin{bmatrix} 2 & 6 & 8\\ 0 & 4 & 5 \end{bmatrix}\\ {R}_{1} \div 2 \rightarrow \begin{bmatrix} 1 & 3 & 4\\ 0 & 4 & 5 \end{bmatrix}\\ {R}_{2} \cdot \frac{1}{4} \rightarrow \begin{bmatrix} 1 & 3 & 4\\ 0 & 1 & \frac{5}{4} \end{bmatrix}\\ {R}_{2} \cdot -3 + {R}_{1} \rightarrow \begin{bmatrix} 1 & 0 & \frac{1}{4}\\ 0 & 1 & \frac{5}{4} \end{bmatrix}\end{align*}$
::[045268] 抽动行[268045]R12}[134045]R2}14}[1340154]R2}3+R1}[10140154]

Example 4
::例4

Reduce the following matrix to row echelon form.
::将下列矩阵减为排梯表。

$\begin{align*}\begin{bmatrix} 3 & 6\\ 2 & 4\\ 5 & 17 \end{bmatrix}\end{align*}$

\begin{aligned} [\begin{matrix} 3 & 6 \\ 2 & 4 \\ 5 & 17 \end{matrix}] \\ R_{1} \div 3, R_{2} \div 2 \to [\begin{matrix} 1 & 2 \\ 1 & 2 \\ 5 & 17 \end{matrix}] \\ R_{1} \cdot - 1 + R_{2} \to [\begin{matrix} 1 & 2 \\ 0 & 0 \\ 5 & 17 \end{matrix}] \\ R_{1} \cdot - 5 + R_{3} \to [\begin{matrix} 1 & 2 \\ 0 & 0 \\ 0 & 7 \end{matrix}] \\ Switch R_{2} and R_{3} \to [\begin{matrix} 1 & 2 \\ 0 & 7 \\ 0 & 0 \end{matrix}] \\ R_{2} \div 7 \to [\begin{matrix} 1 & 2 \\ 0 & 1 \\ 0 & 0 \end{matrix}] \\ R_{2} \cdot - 2 + R_{1} \to [\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}] \end{aligned}

$\begin{align*}\begin{bmatrix} 3 & 6\\ 2& 4\\ 5 & 17 \end{bmatrix}\\ {R}_{1} \div 3, \ {R}_{2} \div 2 \rightarrow \begin{bmatrix} 1 & 2\\ 1 & 2\\ 5 & 17 \end{bmatrix}\\ {R}_{1} \cdot -1 + {R}_{2} \rightarrow \begin{bmatrix} 1 & 2\\ 0 & 0\\ 5 & 17 \end{bmatrix}\\ {R}_{1} \cdot -5 + {R}_{3} \rightarrow \begin{bmatrix} 1 & 2\\ 0 & 0\\ 0 & 7 \end{bmatrix}\\ \text{Switch} \ {R}_{2} \ \text{and} \ {R}_{3} \rightarrow \begin{bmatrix} 1 & 2\\ 0 & 7\\ 0 & 0 \end{bmatrix}\\ {R}_{2} \div 7 \rightarrow \begin{bmatrix} 1 & 2\\ 0 & 1\\ 0 & 0 \end{bmatrix}\\ {R}_{2} \cdot -2 + {R}_{1} \rightarrow \begin{bmatrix} 1 & 0\\ 0 & 1\\ 0 & 0 \end{bmatrix}\end{align*}$
::[3624517]R13、R22[1212517]R11+R2}[1200517]R15+R3}[120007]Switch R2和R3}[120700]R2][7}[120100]R22+R1}[100]

Example 5
::例5

Reduce the following matrix to reduced row echelon form.
::将下列矩阵减为减排梯层表。

$\begin{align*}\begin{bmatrix} 3 & 4 & 1 & 0\\ 5 & -1 & 0 & 1 \end{bmatrix}\end{align*}$

\begin{aligned} [\begin{matrix} 3 & 4 & 1 & 0 \\ 5 & - 1 & 0 & 1 \end{matrix}] \\ R_{1} \cdot 5 \to [\begin{matrix} 15 & 20 & 5 & 0 \\ 5 & - 1 & 0 & 1 \end{matrix}] \\ R_{2} \cdot 3 \to [\begin{matrix} 15 & 20 & 5 & 0 \\ 15 & - 3 & 0 & 3 \end{matrix}] \\ R_{2} - R_{1} \to [\begin{matrix} 15 & 20 & 5 & 0 \\ 0 & - 23 & - 5 & - 3 \end{matrix}] \\ R_{1} \div 15 \to [\begin{matrix} 1 & \frac{4}{3} & \frac{1}{3} & 0 \\ 0 & - 23 & - 5 & 3 \end{matrix}] \\ R_{2} \div - 23 \to [\begin{matrix} 1 & \frac{4}{3} & \frac{1}{3} & 0 \\ 0 & 1 & \frac{5}{23} & - \frac{3}{23} \end{matrix}] \\ R_{2} \cdot - \frac{4}{3} + R_{1} \to [\begin{matrix} 1 & 0 & \frac{1}{23} & \frac{4}{23} \\ 0 & 1 & \frac{5}{23} & - \frac{3}{23} \end{matrix}] \end{aligned}

$\begin{align*}\begin{bmatrix} 3 & 4 & 1 & 0\\ 5 & -1 & 0 & 1\end{bmatrix}\\ {R}_{1} \cdot 5 \rightarrow \begin{bmatrix}15 & 20 & 5 & 0\\ 5 & -1 & 0 &1 \end{bmatrix}\\ {R}_{2} \cdot 3 \rightarrow \begin{bmatrix} 15 & 20 & 5 & 0\\ 15 & -3 & 0 & 3 \end{bmatrix}\\ {R}_{2} - {R}_{1} \rightarrow \begin{bmatrix} 15 & 20 & 5 & 0\\ 0 & -23 & -5 & -3 \end{bmatrix}\\ {R}_{1} \div 15 \rightarrow \begin{bmatrix} 1 & \frac{4}{3} & \frac{1}{3} & 0\\ 0 & -23 & -5 & 3 \end{bmatrix}\\ {R}_{2} \div -23 \rightarrow \begin{bmatrix} 1 & \frac{4}{3} & \frac{1}{3} & 0\\ 0 & 1 & \frac{5}{23} & -\frac{3}{23} \end{bmatrix}\\ {R}_{2} \cdot -\frac{4}{3} + {R}_{1} \rightarrow \begin{bmatrix} 1 & 0 & \frac{1}{23} & \frac{4}{23}\\ 0 & 1 & \frac{5}{23} & -\frac{3}{23} \end{bmatrix}\end{align*}$
::[34105-101]R1}5}[155015-303]R2}[15205015-303]R2—R1}[1520500-23-5-3]R1}[1520500-23-5-3]R1}15}[1431300-23-853]R2}[14313001523-323]R2}[14313001523-323]R2}{43+R1}[1012342301523-323]

Notice how fractions were avoided until the final step. Adding and subtracting large numbers in a matrix is easier to handle than adding and subtracting small numbers because then you don’t need to find a common denominator.
::提醒如何在最后一步之前避免分数。在矩阵中添加和减去大数比增加和减去小数更容易处理,因为那样就不需要找到共同的分母了。

Summary

There are three row operations permitted on matrices: adding a multiple of one row to another, scaling a row by a non-zero constant, and swapping two rows.
::矩阵上允许进行三行操作:在另一行增加一行的倍数,以非零常数缩放一行,并互换两行。

Row echelon form must meet three requirements: leading coefficient of each row is one, all entries in a column below a leading one are zero, and all zero rows are at the bottom of the matrix.
::排梯表必须满足三个要求:每一行的主要系数为一,前列一下的所有条目为零,所有零行都在矩阵的底部。

Reduced row echelon form has an additional requirement: every leading coefficient of 1 must be the only non-zero element in that column.
::减少的排梯表还有一项额外要求:每一主要系数1必须是该栏中唯一的非零要素。

Row reducing a matrix to become the identity matrix illustrates the fact that the rows of the original matrix are linearly independent.
::各行缩小矩阵,成为身份矩阵,说明原始矩阵各行是线性独立的。

Review
::回顾

1. Give an example of a matrix in row echelon form.
::1. 举行梯层表矩阵为例。

2. Give an example of a matrix in reduced row echelon form.
::2. 举一个缩排梯层表矩阵的例子。

3. What are the three row operations you are allowed to perform when reducing a matrix?
::3. 在减少矩阵时,允许执行的三行操作是什么?

4. If a square matrix reduces to the identity matrix, what does that mean about the rows of the original matrix?
::4. 如果一个平方矩阵缩减为身份矩阵,这对原始矩阵的行意味着什么?

Use the following matrix for 5-6.
::5-6使用以下矩阵。

$\begin{align*}A=\begin{bmatrix} -3 & -4 & -12\\ 4 & 4 & 12\\ -11 & -12 & -35 \end{bmatrix}\end{align*}$
::A=[-3-4-4-124412-11-12-35]

5. Reduce matrix $\begin{align*}A\end{align*}$ to row echelon form.
::5. 将表格A到排层表格减少。

6. Reduce matrix $\begin{align*}A\end{align*}$ to reduced row echelon form. Are the rows of matrix $\begin{align*}A\end{align*}$ linearly independent?
::6. 将矩阵A减为排级梯层表:矩阵A各行是否线性独立?

Use the following matrix for 7-8.
::7-8使用以下矩阵。

$\begin{align*}B=\begin{bmatrix} 3 & -4 & 8\\ 9 & 0 & 1\\ 0 & 1 & -2 \end{bmatrix}\end{align*}$
::B=[3-4890101-2]

7. Reduce matrix $\begin{align*}B\end{align*}$ to row echelon form.
::7. 将表B减为排层表。

8. Reduce matrix $\begin{align*}B\end{align*}$ to reduced row echelon form. Are the rows of matrix $\begin{align*}B\end{align*}$ linearly independent?
::8. 将矩阵B减为排级梯层表:矩阵B的行线性独立吗?

Use the following matrix for 9-10.
::9-10使用以下矩阵。

$\begin{align*}C=\begin{bmatrix} 0 & 0 & -1 & -1\\ 3 & 6 & -3 & 1\\ 6 & 12 & -7 & 0 \end{bmatrix}\end{align*}$
::C=[00-1-136-31612-70]

9. Reduce matrix $\begin{align*}C\end{align*}$ to row echelon form.
::9. 将矩阵C减为排行梯层表。

10. Reduce matrix $\begin{align*}C\end{align*}$ to reduced row echelon form. Are the rows of matrix $\begin{align*}C\end{align*}$ linearly independent?
::10. 将矩阵C减为排排梯层表:矩阵C的行线性独立吗?

Use the following matrix for 11-12.
::11-12 使用以下矩阵。

$\begin{align*}D=\begin{bmatrix} 1 & 1\\ 3 & 4\\ 2 & 3 \end{bmatrix}\end{align*}$
::D=[113423]

11. Reduce matrix $\begin{align*}D\end{align*}$ to row echelon form.
::11. 将表格D减为排排表。

12. Reduce matrix $\begin{align*}D\end{align*}$ to reduced row echelon form. Are the rows of matrix $\begin{align*}D\end{align*}$ linearly independent?
::12. 将矩阵D减为行梯层表:矩阵D的行线性独立吗?

Use the following matrix for 13-14.
::13-14使用以下矩阵。

$\begin{align*}E=\begin{bmatrix} -5 & -6 & -12\\ -1 & -1 & -2\\ 2 & 2 & 4 \end{bmatrix}\end{align*}$
::E=[-5-6-6 - 12 - 1 - 1 - 1 - 2224]

13. Reduce matrix $\begin{align*}E\end{align*}$ to row echelon form.
::13. 将表E减为表E。

14. Reduce matrix $\begin{align*}E\end{align*}$ to reduced row echelon form. Are the rows of matrix $\begin{align*}E\end{align*}$ linearly independent?
::14. 将矩阵E减为排级梯层表:矩阵E的行线性独立吗?

Use the following matrix for 15-16.
::15-16使用以下矩阵。

$\begin{align*}F=\begin{bmatrix} -23 & 6 & 3\\ 2 & -\frac{1}{2} & 0\\ -8 & 2 & 1 \end{bmatrix}\end{align*}$
::F=[-23632-120-821]

15. Reduce matrix $\begin{align*}F\end{align*}$ to row echelon form.
::15. 将表格F减为排排表。

16. Reduce matrix $\begin{align*}F\end{align*}$ to reduced row echelon form. Are the rows of matrix $\begin{align*}F\end{align*}$ linearly independent?
::16. 将矩阵F减为排排梯层表:矩阵F各行是否线性独立?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

行操作和行梯表

Row Operations and Row Echelon Forms ::行操作和行梯表

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)