SLT.sagews

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<div class="mathbook-content"><section class="frontmatter" id="frontmatter-1"></section></div>

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<h2 class="heading">
<span class="title">Sage and Linear Algebra Worksheet:</span> <span class="subtitle">FCLA Section SLT</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">Surjective Linear Transformations</span>
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<div class="mathbook-content"><p id="p-1">Two carefully-crafted linear transformations: <code class="code-inline tex2jax_ignore">T</code> is surjective, <code class="code-inline tex2jax_ignore">S</code> is not.</p></div>

A = matrix(QQ, [[2, 2, 5, -2], [2, 3, 1, -4], [-3, -4, -4, 5]])
T = linear_transformation(QQ^4, QQ^3, A, side='right')

T.is_surjective()

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<div class="mathbook-content"><p id="p-2">The range is known in Sage as the “image.”  For a surjective linear transformation, it will be the entire codomain.  Note that the image is a vector space.</p></div>

T.image()

T.image() == T.codomain()

B = matrix(QQ, [[1, -2, 0, 3], [3, -5, 1, 7], [-1, 4, 2, -7]])
S = linear_transformation(QQ^4, QQ^3, B, side='right')

S.is_surjective()

IM = S.image()
IM

IM == S.codomain()

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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">Pre-Images</span>
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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-3">We can create inputs associated with any output.  First, we make an arbitrary output, but make sure it really is an output, as a linear combination of a basis of the image (see basis above).  We print the two vectors in the opposite of what we would consider the “normal” order.</p></article></div>

bas = IM.basis()
out = ()*bas[0] + ()*bas[1]
inp = S.preimage_representative(out)
out, inp

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<div class="mathbook-content"><p id="p-4">A check on our work.</p></div>

S(inp)

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<div class="mathbook-content"><article class="exercise-like" id="exercise-2"><h6 class="heading">
<span class="type">Demonstration</span> <span class="codenumber">2</span>.</h6>
<p id="p-5">We can make other inputs, using the kernel.</p>
<p id="p-6">Any value of <code class="code-inline tex2jax_ignore">new_inp</code> is in the preimage of <code class="code-inline tex2jax_ignore">out</code>, and every element of the preimage can be built this way.  Notice the role the kernel plays, much like in the worksheet about injective linear transformations.</p></article></div>

K = S.kernel()
K

z = K.random_element()
new_inp = inp + z
new_inp, S(new_inp)

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<div class="mathbook-content"><article class="exercise-like" id="exercise-3"><h6 class="heading">
<span class="type">Demonstration</span> <span class="codenumber">3</span>.</h6>
<p id="p-7">Elements outside the range (image) will have empty preimages.  We mildly “wreck” an element of the range.</p>
<p id="p-8">With two initial entries determined by the zeros and ones in the basis vectors, the third entry must be determined, so we can “twiddle” it just a bit to obtain a vector of the codomain that lies outside the range.  We will ask Sage for a pre-image representative anyway and see what happens.</p></article></div>

in_range = ()*bas[0] + ()*bas[1]
in_range

outside_range = vector(QQ, [ , , ])
S.preimage_representative(outside_range)

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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-9">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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