Linear Algebra

62 Chapter One. Linear SystemsTopic: Computer Algebra SystemsThe linear systems in this chapter are small enough that their solution by handis easy. But large systems are easiest, and safest, to do on a computer. Thereare special purpose programs such as LINPACK for this job. Another populartool is a general purpose computer algebra system, including both commercialpackages such as Maple, Mathematica, or MATLAB, or free packages such asSciLab,, MuPAD, or Octave.For example, in the Topic on Networks, we need to solve this.i 0 − i 1 − i 2 = 0i 1 − i 3 − i 5 = 0i 2 − i 4 + i 5 = 0i 3 + i 4 − i 6 = 05i 1 + 10i 3 = 102i 2 + 4i 4 = 105i 1 − 2i 2 + 50i 5 = 0It can be done by hand, but it would take a while and be error-prone. Using acomputer is better.We illustrate by solving that system under Maple (for another system, auser’s manual would obviously detail the exact syntax needed). The array ofcoefficients can be entered in this way> A:=array( [[1,-1,-1,0,0,0,0],[0,1,0,-1,0,-1,0],[0,0,1,0,-1,1,0],[0,0,0,1,1,0,-1],[0,5,0,10,0,0,0],[0,0,2,0,4,0,0],[0,5,-1,0,0,10,0]] );(putting the rows on separate lines is not necessary, but is done for clarity).The vector of constants is entered similarly.> u:=array( [0,0,0,0,10,10,0] );Then the system is solved, like magic.> linsolve(A,u);7 2 5 2 5 7[ -, -, -, -, -, 0, - ]3 3 3 3 3 3Systems with infinitely many solutions are solved in the same way — the computersimply returns a parametrization.ExercisesAnswers for this Topic use Maple as the computer algebra system. In particular,all of these were tested on Maple V running under MS-DOS NT version 4.0. (Onall of them, the preliminary command to load the linear algebra package along withMaple’s responses to the Enter key, have been omitted.) Other systems have similarcommands.