Harvard University,FAS
Fall 2003

Mathematics Math21b
Fall 2003

Linear Algebra
and Differential Equations

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu
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Linear algebra with CAS

We do not assign problems which need computer algebra software (CAS) in this course. Nevertheless, it is useful to know what can be done with such programs, the four M's. We list some Mathematica, Matlab or Maple and Maxima commands which should speak for themselves.


Mathematica

Harvard has a Mathematica site license. You can get it here and request a password, using the Harvard Site License Number L2482-2405.

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
A\v
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);


Please send comments to knill@math.harvard.edu


Fri Jan 30 20:21:17 EST 2004