Sage and Linear Algebra Worksheet

Sage and Linear Algebra Worksheet: FCLA Section VR

Robert Beezer

Department of Mathematics and Computer Science
University of Puget Sound

Fall 2019

Section 1 Vector Representations

It is easy to form vector representations of vectors in \(\mathbb{C}^n\text{.}\)

We get a nonstandard basis quickly from the columns of a nonsingular matrix. The keyword algorithm='unimodular' requests a matrix with determinant \(1\text{.}\)

The columns of A become the “user basis” of a vector space.

Now, we get values of the invertible linear transformation \(\rho_B\) with the Sage method .coordinate_vector() method of the vector space.

The inverse linear transformation is also available as the .linear_combination_of_basis() method of the vector space.

And the automated check:

Notice that this is something we could do “by hand” with just reduced row-echelon form. The coordinitization of u relative to the basis B is just a (unique) solution to a linear system.

The following stanza will always return True as we “coordinatize” and then use the coordinates to form a linear combination of the basis.

Section 2 Abstract Vector Spaces

Sage does not implement abstract vector spaces. It presumes we have “nice” standard bases available and can apply an intermediate coordinatization ourselves.

Demonstration 1.

In \(P_3\text{,}\) the vector space of polynomials with degree at most \(3\text{,}\) find the vector representation of \(p = x^{3} + x^{2} + \frac{1}{2} \, x - \frac{33}{14}\) relative to the basis for \(P_3\text{:}\)

\begin{align*} B = \{& 5x^{3} + 2x^{2} + x + 1,\, -8x^{3} - 3x^{2} - x - 2,\\ & 7x^{3} + 4x^{2} + x + 2,\, -7x^{3} + 3x^{2} + x - 2\}\text{.} \end{align*}

Hint: Coordinatize with respect to the basis \(\left\{1, x, x^2, x^3\right\}\text{.}\)

B is a basis, since A is nonsingular.

Now coordinatize p.

We'll get a coordinatization old-style.

Let's check to see if this is right and we can recover p.