<HTML>
<HEAD>
<TITLE>Linear Algebra -- Exam 1 Review</TITLE>
</HEAD>

<BODY BGCOLOR="#FFFFFF">

<CENTER><H3>Mathematics 44, section 1 -- Linear Algebra</H3></CENTER>
<CENTER><H3>Review Sheet for Exam 1</H3></CENTER>

<P>
<H3>General Information</H3>

<P>
As announced in the course syllabus, the first exam of the semester will 
be given in class on Friday, February 19.  You will have the full class 
period to work on the exam. The format will be similar to that of the 
exams from Algebraic Structures last semester -- four or five problems, 
each possibly containing several parts.  One question this time may 
consist of several ``true - false'' questions where you must either
give a short proof (if you think the statement is true), or a counterexample
(if you think the statement is false).

<H3>Topics to be Covered</H3>

<P>
This exam will cover all the material since the start of the semester, 
through and including the dimension of a vector space and the examples
from class on February 12.  (This is Chapters 1 - 7 in Smith.)
Here is a specific list:

<OL>
<LI>  The axioms for vector spaces -- showing that a set is or is not
a vector space using the axioms
<LI>  The key examples of vector spaces:  <EM><B>R</B><SUP>k</SUP></EM>,
<EM>P<SUB>k</SUB>(<B>R</B>)<EM>, <EM>Fun(S)</EM> and their properties
<LI>  Subspaces of a vector space -- know how to show that a subset
of a vector space is or is not a subspace using the definition (see page 35)
<LI>  Linear combinations and the linear span of a set of vectors
<LI>  Linear dependence and independence
<LI>  Bases of a vector space and the dimension of a vector space
</OL>

<P>
<H3>Proofs to Know</H3>

<P>
<OL>
<LI>  The linear span of any set <EM>E</EM> in a vector space <EM>V</EM>
is a vector subspace of <EM>V</EM>.
<LI>  A set <EM>E</EM> is linearly dependent if and only if
there exists some <EM>A</EM> in <EM>E</EM> such that <EM>A</EM>
is linearly dependent on <EM>E - {A}</EM> (and if so,
<EM>L(E) = L(E - {A})</EM>.
<LI>  If <EM>V</EM> is a vector space with some finite spanning set, 
then every two bases of <EM>V</EM> have the same number of elements.
(You may state <EM>without proof</EM> any facts about solutions of systems
of linear equations that you need here.)
</OL>

<P>
<H3>Review Problems</EM>

<P>
From Smith:  Chapter 2/12,13; Chapter 3/5,6,8 (the notation is defined in
Example 4 on page 29); Chapter 4/7,8,16-20,31; Chapter 5/6,17,18,19 (assume
that the vectors are linearly independent as in 18), Chapter 6/1 - 7, 9;
Chapter 7/1, 5, 9.
</BODY>
</HTML>
