Final Exam
The final for this course will be on , from 6:30 to 8:30 pm.
It will be held in Olin 107
Topics covered:
* all material on eigenvalues and eigenvectors. Including application to dynamical systems. See class notes from the week Feb 16-20
* notion of a basis and coordinates relative to a basis
* matrix algebra
especially powers, because they relate to solutions of dynamical systems
also: limits of powers of matrices
orthogonal basis
* solutions of systems
Gauss Jordan algorithm
assigning arbitrary variables, vector form of solutions. Computing eigenvectors
PROOFS it might be a good idea to know:
1. if A = P D P-1 then Ak = P DkP-1
2. if { u1,u2, . . . ,un} is an orthogonal basis for Rn then the coordinates of any vector x relative to this basis are given by:
  ci = xui / uiui   for i = 1 . . . n  (dot products)
3. (AB)-1 = B-1A-1
4.Det(A) = 0 => the system Ax = 0 has nontrivial solutions.
5. the solution to the dynamical system x(k+1) = Ax(k),   x(0)=x0 is given by:
 x(k) =Akx0
6.If A is diagonalizable and all its eigenvalues are less than one in absolute value then
 lim Ak = 0 as k -> infinity