Math21b, Spring 2016, Linear Algebra and Differential Equations,

The Mathematica lab is due on April 29 (post stamp of email needs to be arrive Friday April 29). We had a workshop on Tue April 26, 7PM -8 PM in Hall B.

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

Mathematica

get it here and request a password,

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);

To fit data with Mathematica, you can use either the built in routines

       data={{4,5},{2,10},{1,100},{5,3}};functions={1,x,Sin[x]};
       Fit[data,functions,x]

or crank in the linear algebra: a+bx+c sin[x] =y

       A=N[{{1,4,Sin[4]},{1,2,Sin[2]},{1,1,Sin[1]},{1,5,Sin[5]}}]; b={5,10,100,3};
       Inverse[Transpose[A].A].Transpose[A].b

With both approaches you get in this example the function 210.3-60x-77.2 Sin[x] which beset fits the data points (4,5),(2,10),(1,100),(5,3).