Linear Algebra

Linear Systems
Summary

Vector Spaces
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Maps Between Spaces
Summary

Determinants
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Similarity
Summary

Linear Systems

Solving Linear Systems
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Linear Geometry of n-Space
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Reduced Echelon Form
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Topics about Linear Systems
Summary

Vector Spaces

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Maps Between Spaces

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Determinants

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Similarity

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Solving Linear Systems

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.

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.

Since the sum of moments on the left of each balance equals the sum of moments on the right (the moment of an object is its mass times its distance from the balance point), the two balances give this system of two equations.

The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene C₇H₈ and nitric acid HNO₃ to produce trinitrotoluene C₇H₅O₆N₃ 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

$x,{rm C}_7{rm H}_8 + y,{rm H}{rm N}{rm O}_3 quadlongrightarrowquad z,{rm C}_7{rm H}_5{rm O}_6{rm N}_3 + w,{rm H}_2{rm O}$

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

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

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.

Gauss' Method
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Describing the Solution Set
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General = Particular + Homogenous
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Comparing Set Descriptions
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Automation
Summary

Linear Geometry of n-Space

Content

Reduced Echelon Form

Content

Topics about Linear Systems

Computer Algebra Systems
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Input-Output Analysis
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Accuracy of Computations
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Analyzing Networks
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Speed of Gauss' Method
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Computer Algebra Systems

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Input-Output Analysis

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Accuracy of Computations

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Analyzing Networks

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Speed of Gauss' Method

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Gauss' Method

Definition 1.1

linear equation

in variables

$x_1,x_2,ldots,x_n$

has the form

$a_1x_1+a_2x_2+a_3x_3+cdots+a_nx_n=d$
where the numbers $a_1, ldots ,a_ninmathbb{R}$ are the equation's coefficients and $dinmathbb{R}$ is the constant. An $n$ -tuple $(s_1,s_2,ldots ,s_n)inmathbb{R}^n$ is a solution of, or satisfies, that equation if substituting the numbers $s_1$ , ..., $s_n$ for the variables gives a true statement: $a_1s_1+a_2s_2+ldots+a_ns_n=d$ .

A system of linear equations

$begin{array}{*{4}{rc}r} a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &cdots &+ &a_{1,n}x_n &= &d_1 \ a_{2,1}x_1 &+ &a_{2,2}x_2 &+ &cdots &+ &a_{2,n}x_n &= &d_2 \ & & & & & & &vdots \ a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &cdots &+ &a_{m,n}x_n &= &d_m end{array}$

has the solution

$(s_1,s_2,ldots ,s_n)$

if that

$n$

-tuple is a solution of all of the equations in the system.

Describing the Solution Set

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General = Particular + Homogenous

Content

Comparing Set Descriptions

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Automation

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