线性代数/Solving Linear Systems 求解线性方程组

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Systems of linear equations are common in science and mathematics.
These two examples from high school science Template:Harv
give a sense of how they arise.

线性方程组在科学和数学中很常见。这两个来自《高中科学》（奥南1990）的例子让我们了解了它们是如何产生的。

The first example is from Physics.
Suppose that we are given three objects,
one with a mass known to be 2 kg, and are asked to find the unknown masses.
Suppose further that
experimentation with a meter stick produces these two balances.

第一个例子来自物理学。假设我们得到三个物体，一个质量已知为2公斤，要求出另外两个物体质量。进一步假设，使用天平进行试验可以产生这两种平衡。

Since the sum of magnitudes of the torques of the clockwise forces equal those of the counter clockwise forces (the torque of an object rotating about a fixed origin is the cross product of the force on it and its position vector relative to the origin; gravitational acceleration is uniform we can divide both sides by it). The two balances give this system of two equations.

由于顺时针力的力矩大小之和等于逆时针力的大小（绕固定原点旋转的物体的力矩是其上的力与其相对于原点的位置矢量的叉积；重力加速度是均匀的，我们可以用它来划分两边）。这两个天平给出了这个由两个方程组成的方程组。

40
h
15
c

=
100

25
c

=
50
50
h

{\displaystyle \begin{array}{rl}
40h+15c &= 100 \\
25c &= 50+50h
\end{array}}

{Can you finish the solution?
你能求出解吗？

The second example of a linear system
is from Chemistry.
We can mix, under controlled conditions, toluene

C

7

H

8

{\displaystyle \hbox{C}\_7\hbox{H}\_8}

 and
nitric acid

H

N

O

3

{\displaystyle \hbox{H}\hbox{N}\hbox{O}\_3}

 to produce
trinitrotoluene

C

7

H

5

O

6

N

3

{\displaystyle \hbox{C}\_7\hbox{H}\_5\hbox{O}\_6\hbox{N}\_3}

along with the byproduct water
(conditions have to be controlled very well, indeed— trinitrotoluene
is better known as TNT).
In what proportion should those components be mixed?
The number of atoms of each element present before the reaction

线性系统的第二个例子来自化学。我们可以在受控制的情况下混合甲苯（

C

7

H

8

{\displaystyle \hbox{C}\_7\hbox{H}\_8}

）
和硝酸（

H

N

O

3

{\displaystyle \hbox{H}\hbox{N}\hbox{O}\_3}

）的生产条件三硝基甲苯（

C

7

H

5

O

6

N

3

{\displaystyle \hbox{C}\_7\hbox{H}\_5\hbox{O}\_6\hbox{N}\_3}

）及其副产物水
（条件必须控制得很好，实际上，三硝基甲苯被称为TNT）。这些成分应该按多大比例混合？每种元素在反应前存在的原子数

x

C

7

H

8

y

H

N

O

3

⟶

z

C

7

H

5

O

6

N

3

w

H

2

O

{\displaystyle
x\,{\rm C}\_7{\rm H}\_8\ +\ y\,{\rm H}{\rm N}{\rm O}\_3
\quad\longrightarrow\quad
z\,{\rm C}\_7{\rm H}\_5{\rm O}\_6{\rm N}\_3\ +\ w\,{\rm H}\_2{\rm O}
}

must equal the number present afterward.
Applying that principle to the elements C, H, N, and O in turn gives
this system.

一定与反应后存在的原子数相等。根据这个反应式中的C、H、N和O元素的守恒。

7
x

=
7
z

8
x
1
y

=
5
z
2
w

1
y

=
3
z

3
y

=
6
z
1
w

{\displaystyle \begin{array}{rl}
7x &= 7z \\
8x +1y &= 5z+2w \\
1y &= 3z \\
3y &= 6z+1w
\end{array}}

<quiz display=simple points=1/1>
{Can you balance the equation?

你能配平这个方程吗？
{ 1\_1 }

C

7

H

8

{\displaystyle {\rm C}\_7{\rm H}\_8\ +\ }

{ 3\_1 }

H

N

O

3

⟶

{\displaystyle {\rm H}{\rm N}{\rm O}\_3\quad\longrightarrow\quad}

{ 1\_1 }

C

7

H

5

O

6

N

3

{\displaystyle {\rm C}\_7{\rm H}\_5{\rm O}\_6{\rm N}\_3\ +\ }

{ 3\_1 }

H

2

O

{\displaystyle {\rm H}\_2{\rm O}}

</quiz>
To finish each of these examples requires solving a system of equations.
In each, the equations involve only the first power of the variables.
This chapter shows how to solve any such system.

要完成这些例子中的每一个都需要解一个方程组。在每一个方程中，方程只涉及未知数的一次方。本章介绍如何求解任何此类方程组。

##### References

* Template:Citation.

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