Linear Algebra/Solving Linear Systems 求解線性方程組

Linear Algebra
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Systems of linear equations are common in science and mathematics. These two examples from high school science (O'Nan 1990) give a sense of how they arise.
線性方程組在科學和數學中很常見。這兩個來自《高中科學》（奧南1990）的例子讓我們了解了它們是如何產生的。

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.
第一個例子來自物理學。假設我們得到三個物體，一個質量已知為2公斤，被要求找出未知質量。進一步假設，使用儀表棒進行試驗可以產生這兩種平衡。

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.
由於順時針力的力矩大小之和等於逆時針力的大小（繞固定原點旋轉的物體的力矩是其上的力與其相對於原點的位置向量的叉積；重力加速度是均勻的，我們可以用它來劃分兩邊）。這兩個天平給出了這個由兩個方程組成的系統。

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

The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene ${\hbox{C}}_{7}{\hbox{H}}_{8}$ and nitric acid ${\hbox{H}}{\hbox{N}}{\hbox{O}}_{3}$ to produce trinitrotoluene ${\hbox{C}}_{7}{\hbox{H}}_{5}{\hbox{O}}_{6}{\hbox{N}}_{3}$ 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

線性系統的第二個例子來自化學。我們可以在受控制的情況下混合甲苯（C₇H₈）和硝酸（HNO₃）的生產條件三硝基甲苯（C₇H₅O₆N₃）及其副產水（條件必須控制得很好，實際上，三硝基甲苯被稱為TNT）。這些成分應該按多大比例混合？每種元素在反應前存在的原子數

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.

一定與反應後存在的原子數相等。根據這個反應中的C、H、N和O元素的守恆。

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

{\rm {C}}_{7}{\rm {H}}_{8}\ +\

{\rm {H}}{\rm {N}}{\rm {O}}_{3}\quad \longrightarrow \quad

{\rm {C}}_{7}{\rm {H}}_{5}{\rm {O}}_{6}{\rm {N}}_{3}\ +\

{\rm {H}}_{2}{\rm {O}}

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.

要完成這些例子中的每一個都需要解一個方程組。在每一個方程中，方程只涉及變量的一次方。本章介紹如何解決任何此類系統。

References[編輯]

O'Nan, Micheal (1990), Linear Algebra (3rd ed.), Harcourt College Pub .

Linear Algebra
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