M A T H 2 1 B
Mathematics Math21b Spring 2008
Linear Algebra and Differential Equations
CAS
Office: SciCtr 434

It is useful to know what can be done with computer algebra systems "the four M's". Mathematica, Matlab or Maple and Maxima. The example snippets should become selfexplanatory during the course.

## Mathematica

Harvard has a Mathematica site license. You can get it here and request a password, using the Harvard Site License Number L2983-5986 (L2482-2405 for faculty staff).

 A={{1,2,3},{4,5,5},{6,7,8}} v={5,-2,3} Inverse[A] A.v A.A.A LinearSolve[A,v] RowReduce[A] QRDecomposition[{{1,0,0},{1,1,0},{1,1,1}}] Fit[{{0,0},{0,1},{1,3}},{1,x,x^2},x] CharacteristicPolynomial[A,x] Tr[A] Det[A] Eigenvalues[A] Eigensystem[A]

## Matlab

Matlab is a CAS which is strong in linear algebra. Matlab is available as a student version. Here are some of the above commands in Matlab.

 A = [1 2 3; 4 5 5; 6 7 8] v = [5;-2;3] inv(A) A*v A*A*A Av rref(A) qr(A) poly(A) det(A) trace(A) eig(A) [v,d]=eig(A)

## Maple

Maple is a CAS comparable with Mathematica or Matlab. Here are the same commands in the Maple dialect.

 with(linalg); A:=[[1,2,3],[4,5,5],[6,7,8]]; v:=[5,-2,3]; inverse(A); multiply(A,v); evalm(A*A*A); linsolve(A,v); rref(A); v1:=[1,0,0]; v2:=[1,1,0]; v3:=[1,1,1]; GramSchmidt({v1,v2,v3}); charpoly(A,x); trace(A); det(A); eigenvalues(A); eigenvectors(A);

## Maxima

Maxima is an open source CAS originally developed by the DOE. While having less features than the commercial CAS, it is GPL'd and free software: you can see the code.
(echelon(A) is here an upper triangular matrix);
 A: matrix([1,2,3],[4,5,5],[6,7,8]); v: [5,-2,3]; invert(A); A.v; A.A.A; linsolve([x+z=5,x+5*y=-2,x-z=0],[x,y,z]); echelon(A); load(eigen); gramschmidt(A); determinant(A); charpoly(A,x); eigenvalues(A); eigenvectors(A);