MATH
21 B
Mathematics Math21b Spring 2017
Linear Algebra and Differential Equations
CAS
Course Head: Oliver Knill
Office: SciCtr 432



The Mathematica lab is due on April 28 (post stamp of email needs to be arrive Friday April 28). We had a workshop on Wed April 26, 6PM -7 PM in Hall C.
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).


Please send questions and comments to knill@math.harvard.edu
Math21b Harvard College Course ID:110989| Oliver Knill | Spring 2017 | Department of Mathematics | Faculty of Art and Sciences | Harvard University, [Canvas, for admin], Twitter