VR.sagews

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<h2 class="heading">
<span class="title">Sage and Linear Algebra Worksheet:</span> <span class="subtitle">FCLA Section VR</span>
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<div class="author">
<div class="author-name">Robert Beezer</div>
<div class="author-info">Department of Mathematics and Computer Science<br>University of Puget Sound</div>
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<div class="date">Fall 2019</div>
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<div class="mathbook-content"><section class="section" id="section-1"><h6 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">1</span> <span class="title">Vector Representations</span>
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<div class="mathbook-content"><p id="p-1">It is easy to form vector representations of vectors in \(\mathbb{C}^n\text{.}\)</p></div>

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<div class="mathbook-content"><p id="p-2">We get a nonstandard basis quickly from the columns of a nonsingular matrix.  The keyword <code class="code-inline tex2jax_ignore">algorithm='unimodular'</code> requests a matrix with determinant \(1\text{.}\)</p></div>

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

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<div class="mathbook-content"><p id="p-3">The columns of <code class="code-inline tex2jax_ignore">A</code> become the “user basis” of a vector space.</p></div>

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

u = random_vector(QQ, n)
u

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<div class="mathbook-content"><p id="p-4">Now, we get values of the invertible linear transformation \(\rho_B\) with the Sage method <code class="code-inline tex2jax_ignore">.coordinate_vector()</code> method of the vector space.</p></div>

c = V.coordinate_vector(u)
c

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<div class="mathbook-content"><p id="p-5">The inverse linear transformation is also available as the <code class="code-inline tex2jax_ignore">.linear_combination_of_basis()</code> method of the vector space.</p></div>

round_trip = V.linear_combination_of_basis(c)
round_trip

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<div class="mathbook-content"><p id="p-6">And the automated check:</p></div>

u == round_trip

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<div class="mathbook-content"><p id="p-7">Notice that this is something we could do “by hand” with just reduced row-echelon form. The coordinitization of <code class="code-inline tex2jax_ignore">u</code> relative to the basis <code class="code-inline tex2jax_ignore">B</code> is just a (unique) solution to a linear system.</p></div>

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

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<div class="mathbook-content"><p id="p-8">The following stanza will always return <code class="code-inline tex2jax_ignore">True</code> as we “coordinatize” and then use the coordinates to form a linear combination of the basis.</p></div>

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

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<div class="mathbook-content"><section class="section" id="section-2"><h6 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">2</span> <span class="title">Abstract Vector Spaces</span>
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<div class="mathbook-content"><p id="p-9">Sage does not implement abstract vector spaces.  It presumes we have “nice” standard bases available and can apply an intermediate coordinatization ourselves.</p></div>

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<div class="mathbook-content"><article class="exercise-like" id="exercise-1"><h6 class="heading">
<span class="type">Demonstration</span> <span class="codenumber">1</span>.</h6>
<p id="p-10">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 \(P_3\text{:}\)</p>
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\begin{align*}
B = \{&amp;
5x^{3} + 2x^{2} + x + 1,\,
-8x^{3} - 3x^{2} - x - 2,\\
&amp; 7x^{3} + 4x^{2} + x + 2,\,
-7x^{3} + 3x^{2} + x - 2\}\text{.}
\end{align*}
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<p id="p-11">Hint:  Coordinatize with respect to the basis \(\left\{1, x, x^2, x^3\right\}\text{.}\)</p></article></div>

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

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<div class="mathbook-content"><p id="p-12"><code class="code-inline tex2jax_ignore">B</code> is a basis, since <code class="code-inline tex2jax_ignore">A</code> is nonsingular.</p></div>

A.is_singular()

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<div class="mathbook-content"><p id="p-13">Now coordinatize <code class="code-inline tex2jax_ignore">p</code>.</p></div>

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

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<div class="mathbook-content"><p id="p-14">We'll get a coordinatization old-style.</p></div>

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

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<div class="mathbook-content"><p id="p-15">Let's check to see if this is right and we can recover <code class="code-inline tex2jax_ignore">p</code>.</p></div>

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

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<div class="mathbook-content"><article class="conclusion" id="conclusion-1"><h5 class="heading"><span></span></h5></article></div>

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<div class="mathbook-content"><p id="p-16">This work is Copyright 2016–2019 by Robert A. Beezer.  It is licensed under a <a class="external" href="https://creativecommons.org/licenses/by-sa/4.0/" target="_blank">Creative Commons Attribution-ShareAlike 4.0 International License</a>.</p></div>

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