The Mathematica lab is now available. You can download it
here [lab.nb]. You can preview a
PDF version of the lab here.
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]
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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)
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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);
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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);
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