VR.ipynb

Kernel:

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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

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 $1\text{.}$

n = 6
A = random_matrix(QQ, n, algorithm='unimodular', upper_bound=9)
A

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

B = A.columns()
V = (QQ^n).subspace_with_basis(B)
V

u = random_vector(QQ, n)
u

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

c = V.coordinate_vector(u)
c

The inverse linear transformation is also available as the .linear\_combination\_of\_basis() method of the vector space.

round_trip = V.linear_combination_of_basis(c)
round_trip

And the automated check:

u == round_trip

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.

aug = column_matrix(B + [u])
aug.rref()

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

w = random_vector(QQ, n)
x = V.coordinate_vector(w)
y = V.linear_combination_of_basis(x)
y == w

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 $\left\{1, x, x^2, x^3\right\}\text{.}$

A = matrix(QQ, [[1, -2, 2, -2], [1, -1, 1, 1], [2, -3, 4, 3], [5, -8, 7, -7]])
B = A.columns()
B

B is a basis, since A is nonsingular.

A.is_singular()

Now coordinatize p.

p = vector(QQ, [-33/14, 1/2, 1, 1])
p

We'll get a coordinatization old-style.

aug = column_matrix(B + [p])
r = aug.rref()
r

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

soln = r.column(4)
round_trip = sum([soln[i]*B[i] for i in range(4)])
round_trip, round_trip == p

This work is Copyright 2016–2019 by Robert A. Beezer. It is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.