Sage and Linear Algebra Worksheet

Sage and Linear Algebra Worksheet: FCLA Section B

Robert Beezer

Department of Mathematics and Computer Science
University of Puget Sound

Fall 2019

Section 1 Bases

Five “random” vectors, each with 4 entries, collected into a set S.

Consider the subspace spanned by these five vectors. We will make these vectors the rows of a matrix and row-reduce to see a basis for the space (subspace, or row space, take your pick). This is an application of Theorem BRS.

Sage does this semi-automatically, tossing zero rows for us.

Demonstration 1.

Construct a random vector, w, in this subspace by choosing scalars for a linear combination of the vectors we used to build W as a span originally.

Then use the three basis vectors in B to recreate the vector w. Question: how many ways can you do this? By Theorem VRRB there should always be exactly one way to create w using a linear combination of a basis of W.

Section 2 Nonsingular Matrices

We will obtain a basis of \(\mathbb{C}^{10}\) from the columns of a \(10\times 10\) nonsingular matrix.

A totally random vector with 10 entries:

Demonstration 2.

By Theorem CNMB, the columns of the matrix are a basis of \(\mathbb{C}^{10}\text{.}\) So the vector v should be a linear combination of the columns of the matrix. Verify this fact in three ways.

First, the old-fashioned way, thus exposing Theorem NMUS.
Then, the modern way, with an inverse, since a nonsingular matrix is invertible, thus exposing Theorem SNCM.
Finally, the Sage way, as described below.

The Sage way: first create a space with a user basis.

Sage still carries an echelonized basis, in addition to the user-installed basis.

Now ask for a coordinatization, relative to the basis in X, thus exposing Theorem VRRB.

Section 3 Orthonormal Bases

A particularly simple orthonormal basis of \(\mathbb{C}^3\text{,}\) collected into the set S.

Demonstration 3.

If these vectors are an orthonormal basis, then as the columns of a matrix they should create an orthonormal basis.

Demonstration 4.

Build a random vector of size \(3\) and find our ways to express the vector as a (unique) linear combination of the basis vectors. Which method is most efficient?

A totally random vector with 3 entries.

First, the old-fashioned way, thus exposing Theorem NMUS.

Now, the modern way, with an inverse, since a nonsingular matrix is invertible, thus exposing Theorem SNCM.

The Sage way. Create a space with a “user basis” and ask for a coordinatization, thus exposing Theorem VRRB.

Finally, exploiting the orthonormal basis, and computing scalars for the linear combination with an inner product, thus exposing Theorem COB. (Sage's .inner_product() does not conjugate the entries of either vector, so we use the more careful .hermitian_inner_product() vector method instead.)