NM.ipynb

Kernel:

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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.

A = random_matrix(QQ, 5, algorithm="unimodular", upper_bound=20)
A

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:

A.is_singular()

Now, a carefully crafted singular matrix.

B = matrix(QQ, [[ 7, -1, -12, 21, -8],
                [-3,  3,   0, -9,  6],
                [ 3,  3, -12,  9,  0],
                [ 2,  3, -10,  6,  1],
                [-2,  2,   0, -6,  4]])
B

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:

B.is_singular()

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

c = vector(QQ, [17,-15,-3,-5,-10])
d = vector(QQ, [-3,1,-2,1,2])

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 in Sage. It's description contains a lot of things we do not understand yet.

NS = B.right_kernel()
NS

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.

u = vector(QQ, [0,0,3,4,6])

v = vector(QQ, [1,0,0,5,-2])

This work is Copyright 2016–2019 by Robert A. Beezer. It is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.