Lecture 12
Gaussian Elimination (with pivoting) as reported
by Chris McAloney.
We began this lecture by discussing some basic
terminology and
properties of matrices and
matrix algebra, with the intended goal
of
outlining some criteria for determining
whether or not a system of
equations has a
solution and, if so, how many solutions it has. The
n
x n identity matrix was defined, as well as
the notion of an
invertible (or non-singular)
matrix. The concept of linear
independence
was briefly outlined, as well as that of the rank of
a
matrix A, and we tied together the concepts
of invertibility, linear
independence, and
matrix rank, since all these notions can be used
to
determine that a system of equations has
exactly one solution.
Then, we moved on
to Gaussian elimination with pivoting. An
example
was shown which demonstrated that the
elimination algorithm presented
in Lecture 11
could fail if, during the course of the algorithm, a
row
operation caused a zero to appear in one
of the pivot elements (the
elements along the
main diagonal) of the coefficient matrix.
The
original algorithm was then modified,
through the addition of row
swapping, to
prevent this occurrence.
Posted: Fri - October 8, 2004 at 10:06 AM