Sage and Linear Algebra Worksheet: FCLA Section IVLT

Section 1 Invertible Linear Transformations

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

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.

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.

Vector spaces with defined user bases.

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

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

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

Section 3 Rank and Nullity

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

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

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

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