ILT.ipynb

Kernel:

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Sage and Linear Algebra Worksheet: FCLA Section ILT

Robert Beezer

Department of Mathematics and Computer Science
University of Puget Sound

Fall 2019

Section 1 Injective Linear Transformations

Two carefully-crafted linear transformations: T is injective, S is not.

A = matrix(QQ, [[1, 2, 2], [3, 7, 6], [1, 2, 1], [2, 5, 7]])
T = linear_transformation(QQ^3, QQ^4, A, side='right')

T.is_injective()

T.kernel()

B = matrix(QQ, [[0, 1, -2], [-1, 1, 3], [-2, 5, 0], [0, 2, -4]])
S = linear_transformation(QQ^3, QQ^4, B, side='right')

S.is_injective()

K = S.kernel()
K

We create two different inputs, which differ by a random vector from the kernel (which we hope is not simply the zero vector, a distinct possibility). We always get the same output from S, predictably. If we try this with T then the kernel vector is always the zero vector and the demonstration is very uninteresting.

z = K.random_element()
u = random_vector(QQ, 3)
w = u + z
u, w, S(u), S(w), S(u) == S(w)

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