Dear 293:       10/2/96

1) HW due Wed 10/9/96 is still not yet composed.
It will come soon. It will cover chapter 4.1 including numerical solution
of systems (trivial extension of stuff you know about ODE23), and a little on solving
systems of two equations as a single second order system.

It will cover some topics with matrices as well.

2) For three lectures 10/3, 10/5, and 10/8 we will be discussing matrices.
A (somewhat artificial) reason for this is to help us with systems of linear ODEs
(Solving [x]' = [A][x]). Another reason is because a little matrix math is
needed soon for some courses  that have  math 293 as a co-requisite (OR 270).  
Finally, a little on matrices is a nice prep for 294 where these topics 
are going to get a little heavy.

Here is what you should know by the end of these three lectures. 
Much but not all of this is covered in the text on pages 
       255-260, 
       268-270, and 
       277-278. 
In the book the matrix topics and ODE topics are intermingled. 
In class we will just do matrices for their own sake for these 3 days.
You might also look in other books, say the math 294 book by Lai (formerly
the 293 book) where these topics are covered in greater detail. 
We will come back to ODEs next week for two lectures.


For the purposes of this note let [A] and [B] be matrices  
                         and  let [x] and [y] column vectors.

Consider this key equation:  [A][x]=[b]  (*)

Know the components aij, xj and bi (i's and j's are subscripts).

Given a system of linear algebraic equations write them in the form (*).

Given the form (*) write the corresponding system of equations.

Given [A] and [x] solve (*) to find [b]. (by hand and MATLAB)

Find the transpose of [A].  (by hand and MATLAB)

Multiply [A][B].  (by hand and MATLAB)

Recognize and define the identity matrix. (create in MATLAB)

Solve (*) for [x] using row operations 
     (by hand for small matrices, with MATLAB for large).

Know what the inverse of a matrix is.
Find the inverse of [A] using row operations 
      (by hand for small matrices, with MATLAB for large).
Use the inverse of [A] to solve * for [x].

Know the definition of an eigenvector and eigenvalue.
   Geometric interpretation.
   Recognize them when  you see them.
   Find them with MATLAB.