**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

**2.** if** { u _{1},u_{2}, . . . ,u_{n}}** is an orthogonal basis for R

c_{i} = **x**u_{i} / u_{i}u_{i} for i = 1 . . . n (dot products)

**3. (AB) ^{-1} = B^{-1}A^{-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)=x _{0}** is given by:

**x(k) =A ^{k}x_{0}**

**6.**If A is diagonalizable and all its eigenvalues are less than one in absolute value then

lim **A ^{k} = 0 ** as k -> infinity