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<div class="mathbook-content"><section class="article" id="ILT"></section></div>
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<h2 class="heading">
<span class="title">Sage and Linear Algebra Worksheet:</span> <span class="subtitle">FCLA Section ILT</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">Injective 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 injective, <code class="code-inline tex2jax_ignore">S</code> is not.</p></div>
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A = matrix(QQ, [[1, 2, 2], [3, 7, 6], [1, 2, 1], [2, 5, 7]])
T = linear_transformation(QQ^3, QQ^4, A, side='right')
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T.is_injective()
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T.kernel()
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B = matrix(QQ, [[0, 1, -2], [-1, 1, 3], [-2, 5, 0], [0, 2, -4]])
S = linear_transformation(QQ^3, QQ^4, B, side='right')
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S.is_injective()
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K = S.kernel()
K
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<div class="mathbook-content"><p id="p-2">We create two different inputs, which differ by a random vector from the kernel (which we hope is not simply the zero vector, a distinct possibility).  We always get the same output from <code class="code-inline tex2jax_ignore">S</code>, predictably.  If we try this with <code class="code-inline tex2jax_ignore">T</code> then the kernel vector is <em class="emphasis">always</em> the zero vector and the demonstration is very uninteresting.</p></div>
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z = K.random_element()
u = random_vector(QQ, 3)
w = u + z
u, w, S(u), S(w), S(u) == S(w)
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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-3">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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