Lecture 10: Linear Systems (Matrix Magic)

Matrices and Linear Systems in Attaway

The following lecture overheads and accompanying reading are independent of MatLab. Attaway Chapter 11.1 links matrices as described here with their representation and use in Matlab. Attaway 11.1 should be read and understood as a complement to the on-line lecture and reading.

Attaway 11.2, leading up to, gives the background and algorithm for Gaussian Elimination as a method of solving Linear Equations. It is useful "how-to" reading for the assignment.

Linear Systems

Geometric Intuition

Matrix Representation

Linear Systems Happen

Gaussian Elimination

Basic Approach

Details A: Reduction Step

Implementation Note: Though it mixes element semantics and is not mathematically a pure object, the augmented matrix of the N-variable system AX = B is the
N x N+1 matrix [A | B] (that is the column of constants B concatenated onto the right of A ). This augmented matrix is useful because all the EROs involve both A and B , which are now together in the augmented matrix and can be manipulated together twice as easily as apart.

Details B: Back Substitution Step

Computational Complexity

Potential Problem A: Zero Pivot

Potential Problem B: Near Zero Pivot

Potential Problem C: Unstable System