The exam will cover chapters 5 and 6 of the book. Let me give you an outline:

We have introduced the notion of a vector space, which can refer to any 
collection of objects with which operations of addition and of multiplication 
by a real number can be defined in such a way that the usual rules known from 
geometric vectors and vectors in R^n apply. The precise list of rules can 
be found in the book and in the glossary. If there were no further examples 
for vector spaces than R^n, the whole wisdom of linear algebra would 
probably not be worthwhile an entire course of its own.

Particular examples of vector spaces other than R^n are; 
- (for any size choice mxn) the vector space of mxn matrices
- (for any m) the vector space of symmetric mxm matrices
- various vector spaces of functions, like (e.g) 
    the vector space of all continuous functions defined for all x in the 
    interval [0,1]

The fundamental notions that apply to any vector space are:
linear combination, spanning set, span, 
linearly dependent versus linearly independent sets, basis, dimension and 
coordinates wrt a basis.

For instance an *example* of a linear combination of the matrices 

[ 1 2 1 ]      [-3 1 7 ]       [ 1 1 6 ]
[ 2 3 5 ]  and [ 1 1 1 ]  and  [-1 2 2 ]
[ 1 0 2 ]      [ 0 2 1 ]       [ 2 1 6 ]

would be 

   [ 1 2 1 ]     [-3 1 7 ]      [ 1 1 6 ]
-1 [ 2 3 5 ] + 3 [ 1 1 1 ]  - 4 [-1 2 2 ]
   [ 1 0 2 ]     [ 0 2 1 ]      [ 2 1 6 ]


an example of a linear combination of the functions f and g 
(where f(x)= cos x and g(x)= x^2)
would be the function 3f-7g given by the formula 3 cos x - 7 x^2

and of course similarly for linear combinations of vectors in R^n.


Make sure you understand the meanings of the other words in the above list
and study the pertinent hwk problems that illustrate them. 

Checking whether a given set of vectors (eg in R^n) is a spanning set for
this vector space, or is  linearly independent, amounts to answering 
questions about linear systems of equations. If you have n vectors in R^n
these questions can be answered by forming a square matrix out of these 
vectors and checking its determinant. However, if you are given, say, 
two vectors in R^3, or four vectors in R^3, determinants do not apply and 
you have to decide with row reduction whether the system of equations has 
a solution for every right hand side, or has only the trivial solution in 
case the right hand side is 0. 

In the case of a vector space of functions you had two problems in the hwk, 
which you should make sure to understand. The book mentions deciding linear 
independence of functions by means of Wronskians, but we did not cover 
this in class.



Given a vector space, finding a basis may be extremely easy, or extremly 
tricky. For instance, it is very easy to give a basis for the vector 
space R^n. In contrast I never asked you (and will never ask you) to give 
a basis for the v.sp. of all continuous functions defined on R. Nobody can do 
this in practice. Manageable problems of finding a basis were #14,#15. They
used insight into the nature of the particular vector space at hand, rather 
than a routine method.

We studied the row, column, and null spaces of a given matrix. For those, 
a routine method to find a basis was explained, and of course you need 
to know it. As a consequence of this method, we learned the notion of a 
rank of a matrix, and the crucial result that
rank A + nullity A = # of columns of A
Obviously you should be able to perform the calculations involved in 
finding such bases, as well as understand the meaning of the notions of 
`row space', `column space' and null space'. 


We finally studied the notion of an inner product in vector spaces. (Ch 6)
You have seen examples and non-examples in R^n, and also an inner product in 
the vector space of nxm matrices, and the inner product in the space of 
continuous functions on [0,1] given by the formula
<f,g> = integral from 0 to 1 f(x) g(x) dx. This is an important inner product
which most of you will encounter in later classes. 
Make sure you understand all these examples.
Also make sure you can use the calculational trick of completing the square
in simple examples where it needs to be determined whether a certain expression 
satisfies the last requirement for an inner product (Ch6, hwk 1b). 

Problem #8,9 in Ch.6 are of paramount importance, but given that you may find it 
not so easy and the time until the exam is short, it will not be required 
on exam 3. Give these two hwk problems due attention first thing after the exam.


An important algorithm is Gram Schmidt orthogonalization. It allows, given 
a basis, to convert this basis into a different one that consists of 
mutually orthogonal vectors. This is important because an orthogonal basis 
is much more convenient in practice. Gram-Schmidt is the last material 
that still goes on exam 3. 

Everything beyond will have to wait for inclusion in the comprehensive final. 

NOTE: Often the last classes before the finals squeeze in material that 
doesn't get digested too well. Take the 3 lessons after exam 3 serious. 
They contain highlights of the course, and stuff you are most likely to need 
later, in particular more about eigenvalues and possibly the QR decomposition
of a matrix..