In this section we define a linear transformation from \(\mathbb{C}^{3}\) to \(\mathbb{C}^{7}\). 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.

Check out a few of these…

Now we build two different representations.

A natural way to build a change-of-basis matrix in Sage is to adjust the bases for the identity linear transformation.

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.

We specialize to linear transformations with equal domain and codomain.

First a matrix representation using a square matrix.

A basis of \(\mathbb{C}^8\). 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.