Subject: Some stuff to study for Exam C.
Date: Thu, 16 Apr 1998 19:25:43 -0400
From: Jonathan King <squash@math.ufl.edu>

Hi Folks
  Here are things to think deeply about before the test.  And naturally,
you should continue to think deeply about them afterwards.  All of this
stuff is stuff that you should know forever!	-Jonathan

Abbreviations:
  def.	definition
  v.s	vectorspace
  trn	transformation
  lin.	linear(ly)

Some Topics to know:
  Everything about DETERMINANTS that we covered (multi-linearity,
homogeneous polynomial, sign of a permutation, geometric interpretation of
the abs.value of det(A) etc.) as well as all the text's stuff 3.1 to 3.3.

  Axioms for a VECTORSPACE and what a SUBSPACE is.  SPANNING SET of
vectors.  LINEARLY INDEPENDENT set of vectors.  Def. of v.s being
FINITE-DIMENSIONAL.  The notion of a BASIS of a finite-dimensional v.s,
and the dimension of such a space.
  Be able to compute by hand (for *small* degrees) the polynomials of
Project Sum and Project Interpolate.  Think about the v.s of polynomials
(E.g, what are its subspaces?  What are their dimensions?  How do I tell
when a set of polynomials is a lin. independent set?)

  The RANGE and KERNEL of a linear trn., the notions of Nul(A), Col(A),
Row(A) and how to compute a basis for them; also, how to determine if a
given vector is in them.  In particular, know sections 4.1-4.6.

  We did the computation of a change-of-basis matrix on Wednesday.
Problems 1-15 on P.270 are good typical problems.   Problem 13 is an
application to the v.s of polynomials, in which we have worked a lot.

  On Friday and Monday, we will do eigenvectors and eigenvalues, and I
will put a/some *simple* eigen-questions on the exam.

  A Typical example: Compute the characteristic poly. of
	A := [2 -3 ; 4 5].
As we will see, the char. poly. of A is simply the determinant of A - xI,
(A minus x times the identity matrix)
that is, the det. of

	2-x	-3
	4	5-x  .

This det. is (2-x)(5-x) - (-3)4, which equals 22 - 7x + x^2.

================================================================

Old Stuff that helps with new stuff: Invertible Matrix Thm.  Knowing rref cold.

================================================================

I will NOT ask you about:
  Iterative solns to Lin.Sys..
  LU-factorizations.
  Geometric interpretation of the sign of det(A).
  Running time of programs.
================================================================

	Note:  As announced, I will be available in my office during 4th
period.  Alas, I will be available only for a few minutes after class.

		Best Wishes,
		  Jonathan