Sage and Linear Algebra Worksheet: FCLA Section CB

Section 1 A Linear Transformation, Two Vector Spaces, Four Bases

In this section we define a linear transformation from \(\mathbb{C}^{3}\) to \(\mathbb{C}^{7}\) using a randomly selected matrix. The definition is a \(7\times 3\) matrix of rank \(3\) that we will use to multiply input vectors with a matrix-vector product. It is not important if the linear transformation is injective and/or surjective.

We will build two representations, using a total of four bases — two for the domain and two for the codomain.

The four bases, associated with the two vector spaces.

Now we build two different representations.

Section 2 Change of Basis Matrices

A natural way to build a change-of-basis matrix in Sage is to adjust the bases for domain and range of the identity linear transformation by supplying an identity matrix to the linear tansformation constructor.

This matrix should convert between the two bases for the domain. Here's a check of Theorem CB.

Same drill in the codomain.

And here is the check on Theorem MRCB. Convert from domain basis 1 to domain basis 2, use the second representation, then convert back from codomain basis 2 to codomain basis 1 and get as a result the representation relative to the first bases.

Section 3 A Diagonal Representation

We specialize to linear transformations with equal domain and codomain.

First a matrix representation using a square matrix.

A basis of \(\mathbb{C}^8\text{.}\) And a vector space with this basis.

That's a nice representation! Where did the basis come from?

Some (right) eigenvectors.

Eigenvalues are a property of the linear transformation.

Bases for the eigenspaces depend on the representation, but the actual eigenvectors are also a property of the linear transformation.

We could do the same thing, but in the style of Section SD, using a change-of-basis matrix.

Here is similarity, in disguise.