IVLT.ipynb

Kernel:

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Sage and Linear Algebra Worksheet: FCLA Section IVLT

Robert Beezer

Department of Mathematics and Computer Science
University of Puget Sound

Fall 2019

Section 1 Invertible Linear Transformations

A carefully-crafted invertible linear transformation from $\mathbb{Q}^5$ to $\mathbb{Q}^5\text{.}$

A = matrix(QQ, [[1, 1, -1, -2, 0], [-3, -2, 1, 4, 7],
    [2, 2, -1, -3, -4], [-4, -3, 3, 8, 3], [5, 6, -7, -8, 8]])
T = linear_transformation(QQ^5, QQ^5, A, side='right')
T

T.is_injective(), T.is_surjective()

T.is_invertible()

S = T.inverse()
S

The \* operator, like we would use for multiplication, will create a composition. This will be perfectly natural once we discuss Section MR. Here, composing an invertible linear transformation with its inverse will yield an identity linear transformation.

comp = S*T
comp

comp.is_identity()

Section 2 Defining an Invertible Linear Transformation on Bases

Now, an invertible linear transformation defined on a basis, and the resulting inverse linear transformation. We get two “random” bases of $\mathbb{Q}^7$ from nonsingular (determinant one) matrices.

C = random_matrix(QQ, 7, 7, algorithm='unimodular', upper_bound=99)
Cbasis = C.columns()
D = random_matrix(QQ, 7, 7, algorithm='unimodular', upper_bound=99)
Dbasis = D.columns()

Vector spaces with defined user bases.

Cspace = (QQ^7).subspace_with_basis(Cbasis)
Dspace = (QQ^7).subspace_with_basis(Dbasis)
Cspace, Dspace

The invertible linear transformation defined with images as the vectors in the codomain basis D.

T = linear_transformation(Cspace, QQ^7, Dbasis)
T

T.is_invertible()

Now we simply “turn around” the definition, to make the inverse.

S = linear_transformation(Dspace, QQ^7, Cbasis)
S

S.is_invertible()

Composition with vector spaces using different bases does not seem to be working properly. So we just check some random inputs to the composition.

comp = S*T
comp.is_identity()

v = random_vector(QQ, 7)
v, T(S(v)) == v, S(T(v)) == v

Section 3 Rank and Nullity

A general (i.e. not invertible) linear transformation from $\mathbb{Q}^6$ to $\mathbb{Q}^5\text{.}$

F = matrix(QQ, [[1, 0, 2, -1, -4, 2], [-1, -1, -4, 3, 6, -5], [0, 1, 3, -2, -4, 5],
    [0, 4, 6, -8, -4, 8], [0, 1, 2, -2, -2, 3]])
R = linear_transformation(QQ^6, QQ^5, F, side='right')
R

Rank is dimension of range (image). Note there are not left/right variants.

R.image()

R.rank()

Nullity is dimension of kernel. Note there are not left/right variants.

R.kernel()

R.nullity()

Note that rank and nullity sum to the dimension of the domain (which is $6$ here).

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