Linear Algebra/Solving Linear Systems

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

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 two balances (one of which is depicted below).

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.


 * $$\begin{array}{rl}

40h+15c &= 100  \\ 25c     &= 50+50h \end{array}$$  {Can you finish the solution? c = { 4_2 } kg h = { 1_2 } kg
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The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene $$\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&mdash; 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.

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.


 * $$\begin{array}{rl}

7x     &= 7z  \\ 8x +1y &= 5z+2w  \\ 1y     &= 3z  \\ 3y     &= 6z+1w \end{array}$$  {Can you balance the equation? { 1_1 }$${\rm C}_7{\rm H}_8\ +\ $${ 3_1 }$${\rm H}{\rm N}{\rm O}_3\quad\longrightarrow\quad$${ 1_1 }$${\rm C}_7{\rm H}_5{\rm O}_6{\rm N}_3\ +\ $${ 3_1 }$${\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.
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