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Sage and Linear Algebra Worksheet: FCLA Section VR
Robert Beezer
Department of Mathematics and Computer Science
University of Puget Sound
Fall 2019
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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.
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.
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.
We'll get a coordinatization old-style.
Let's check to see if this is right and we can recover p.
This work is Copyright 2016–2019 by Robert A. Beezer. It is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.