Sage and Linear Algebra Worksheet

Sage and Linear Algebra Worksheet: FCLA Section NM

Robert Beezer

Department of Mathematics and Computer Science
University of Puget Sound

Fall 2019

First, a guaranteed nonsingular \(5\times 5\) matrix, created at random.

Demonstration 1.

Augment with the zero vector, using the matrix method .augment() and the vector constructor zero_vector(QQ, 5). Then row-reduce to use Definition NM. Or instead, do not augment and apply Theorem NMRRI.

Demonstration 2.

Build some random vectors with random_vector(QQ, 5), augment the matrix and row-reduce. There will always be a unique solution to the linear system represented by the augmented matrix. This is Theorem NMUS.

Instead—cheap, easy and powerful:

Now, a carefully crafted singular matrix.

Demonstration 3.

Augment with the zero vector and row-reduce (Definition NM), or don't augment and row-reduce (Theorem NMRRI).]

Demonstration 4.

A random vector of constants will only rarely build a consistent system when paired with B. Try it. But this is not a theorem, see the vector c below.

Instead—cheap, easy and powerful:

Two carefully crafted vectors for linear systems with B as coefficient matrix.

Demonstration 5.

Which of these two column vectors will create a consistent system for this singular coefficient matrix? (Stay tuned.)

A null space is called a right kernel in Sage. It's description contains a lot of things we do not understand yet.

Demonstration 6.

But we can test membership in the null space, which is the most basic property of a set. Try u in NS and then repeat with v.