Final Exam

The final for this course will be on Tuesday, Feb 24, 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