Lecture 14
Review of Gaussian ELimination LU decomposition
and solutions to linear systems of equations
Today, using an example, I reviewed the
algorithms for Gaussian elimination and LU factorization. We saw that the two
algorithms are essentially one and the same. I did not discuss the issue of
partial pivoting, however, I will continue the same example in a subsequent
lecture to address pivoting.
I also
gave a geometric interpretation of the outcomes that can arise when solving a
system of equations.
That is, the system may
have
1. unique solution,
2. infinitely many solutions,
3. or no
solution.
In two dimensions the
geometric interpretation is
1. unique
solution is the intersection point of two "independent
lines".
2. infinitely many solutions
correspond to all the points on the one line that arises when the system is not
linearly independent but is consistent.
3. or
no solution corresponds to a system of two parallel lines that have no common
intersection point. This is a system that is not linearly independent and not
consistent.
Posted: Thu - October 14, 2004 at 10:19 AM