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<h1>Preface</h1>
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<h2> Contents </h2>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-story">The Story</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#teaching-approach">Teaching Approach</a></li>
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<p><img alt="" src="_images/sophie-germain.jpeg" /></p>
<blockquote>
<div><p>Algebra is but written geometry.</p>
<p>Sophie Germain</p>
</div></blockquote>
<section class="tex2jax_ignore mathjax_ignore" id="preface">
<h1>Preface<a class="headerlink" href="#preface" title="Permalink to this heading">#</a></h1>
<p>Welcome to this book!</p>
<p>These are lecture notes for Computer Science 132, <em>Geometric
Algorithms,</em> as taught by me at Boston University. The overall
structure of the course is roughly based on <em>Linear Algebra and its
Applications,</em> by David C. Lay, Addison-Wesley (Pearson). Many
examples and illustrations are based on that excellent text. However all
the content has been significantly revised by me. The book also
includes contributions and improvements from Wayne Snyder, as well as
from students who have taken CS 132 (see the list of contributors <a class="reference external" href="https://github.com/mcrovella/CS132-Geometric-Algorithms/">here</a>).</p>
<section id="the-story">
<h2>The Story<a class="headerlink" href="#the-story" title="Permalink to this heading">#</a></h2>
<blockquote>
<div><p>I have long embraced the belief that every course should be built
around a story, a quest to answer certain burning questions.</p>
<p>David Bressoud, <a class="reference external" href="https://www.mathvalues.org/masterblog/launchings201906-z45y4">mathvalues.org</a></p>
</div></blockquote>
<!--- Outstanding video on history of related ideas: -->
<!--https://m.youtube.com/watch?v=5M2RWtD4EzI --->
<p>What is linear algebra really <em>about</em>? This is a great question. My
attempt at an answer is this:</p>
<p>Our shared experience of the world is in three dimensions. In that
context humans have acquired innate and learned abilities to think about
shapes and spatial relationships. Linear algebra asks: how would all
that change if <strong>the number of dimensions was unspecified?</strong></p>
<p>In that view, an enormously important contribution comes from an algebra
in which the dimensionality of objects is unspecified so that concepts
become generalized. Another important contribution comes from taking
familiar three-dimensional notions and asking what we can say about them,
and how we can reason about them, in arbitrary dimension.</p>
</section>
<section id="teaching-approach">
<h2>Teaching Approach<a class="headerlink" href="#teaching-approach" title="Permalink to this heading">#</a></h2>
<blockquote>
<div><p>Five-dimensional shapes are hard to visualize – but it doesn’t mean
you can’t think about them. Thinking is really the same as seeing.</p>
<p>WIlliam Thurston</p>
</div></blockquote>
<p>The rationale for the teaching approach used in this course is
<a class="reference external" href="https://github.com/mcrovella/CS132-Geometric-Algorithms/blob/master/Collateral/CS132-Teaching-Philosophy.pdf">here.</a>
In brief:</p>
<p>Students learning Linear Algebra need to develop three modes of
thinking. The first is <em>algebraic</em> thinking – how to correctly manipulate symbols
in a consistent logical framework, for example to solve equations. The
second is <em>geometric</em> thinking:
learning to extend familiar two- and three-dimensional concepts to
higher dimensions in a
rigorous way. The third is <em>computational</em> thinking: understanding the
relationship between abstract algebraic machinery and actual
computations which arrive at the (hopefully) correct answer to a specific problem in
an efficient way.</p>
<blockquote>
<div><p>It’s in words that the magic is — Abracadabra, Open Sesame, and the
rest — but the magic words in one story aren’t magical in the next.
The real magic is to understand which words work, and when, and for
what; the trick is to learn the trick.</p>
<p>John Barth, <em>Chimera</em></p>
</div></blockquote>
<p>Each mode of thinking provides a
distinct, powerful way of understanding a problem, and so
using the full power of linear algebra requires being able to switch between
these modes with fluidity.
However, these three modes of thinking are quite different, and
often students are better at some modes than others. For example, here
are three views of matrix-vector multiplication:</p>
<p><img alt="" src="_images/L0-overview-diagram.jpg" /></p>
<p><a class="reference external" href="https://jupyter.org">Jupyter notebooks</a> – including the use of <a class="reference external" href="https://rise.readthedocs.io/en/stable/">RISE</a> for presentation, Python
for computation, and
<a class="reference external" href="https://jupyterbook.org/intro.html">jupyter books</a> for reference – are
an ideal
teaching environment to take on
this trimodal challenge.
Hence the goal of these notes is to take advantage of
the Jupyter toolchain to interweave these
modes on a fine grain, frequently moving from one mode to the other, to
constantly reinforce connections between ways of thinking about linear algebra.</p>
</section>
<section id="format">
<h2>Format<a class="headerlink" href="#format" title="Permalink to this heading">#</a></h2>
<p>The notes are in the form of Jupyter notebooks. Demos and most figures
are included as executable Python code. All course materials are in
the github repository
<a class="reference external" href="https://github.com/mcrovella/CS132-Geometric-Algorithms">here.</a></p>
<p>Each of the Chapters is based on a single notebook, and each forms the
basis for one lecture (more or less).</p>
<p>I hope you enjoy this book, whose goal is to <strong>prevent</strong> you from falling
into this trap:</p>
<blockquote>
<div><p>Algebra is the offer made by the devil to the mathematician. The devil
says: I will give you this powerful machine, it will answer any
question you like. All you need to do is give me your soul: give up
geometry and you will have this marvelous machine.</p>
<p>Sir Michael Atiyah, 2002</p>
</div></blockquote>
<p>And to conclude, here is an anonymous course review from Fall 2022:</p>
<blockquote>
<div><p>I thought when the Professors were talking about how linear algebra is
“beautiful,” they were exaggerating, but by the end of the course, I
understood why this is true.</p>
</div></blockquote>
</section>
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