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<title>Final Exam Topics</title>
<meta name="author" content="Nathan Mull" />
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<div id="org-div-home-and-up"><a href="material.html">↩</a></div><div id="content" class="content">
<h1 class="title">Final Exam Topics</h1>
<div id="table-of-contents" role="doc-toc">
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#orgbc2d56d">Systems of linear equations</a></li>
<li><a href="#org272a31e">Gaussian elimination</a></li>
<li><a href="#org1e87871">Vector equations</a></li>
<li><a href="#orgb417ad5">Ax = b</a></li>
<li><a href="#orgb050f7e">Linear independence</a></li>
<li><a href="#org8b182ab">Linear transforms</a></li>
<li><a href="#orge59fbae">Matrix of a transformation</a></li>
<li><a href="#orgb61fbed">Matrix algebra</a></li>
<li><a href="#orgfaed484">Matrix inverses</a></li>
<li><a href="#org2d760e2">The invertible matrix theorem</a></li>
<li><a href="#orgfa98bed">Markov chains</a></li>
<li><a href="#org9caaf71">Matrix factorization</a></li>
<li><a href="#org86e368c">Computer graphics</a></li>
<li><a href="#org6c3bd74">Subspaces</a></li>
<li><a href="#org5b8c2dc">Dimension and rank</a></li>
<li><a href="#orgc4ae8b8">Eigenvectors and eigenvalues</a></li>
<li><a href="#org3924a0f">The characteristic equation</a></li>
<li><a href="#org30d6a09">Diagonalizability</a></li>
<li><a href="#orgd2109bd">PageRank</a></li>
<li><a href="#orgad264ae">Analytic geometry</a></li>
<li><a href="#orgcfde3b3">Orthogonal projection</a></li>
<li><a href="#org6a2191c">Least squares</a></li>
<li><a href="#orgd5c348f">Linear models</a></li>
<li><a href="#org43b8af7">Symmetric matrices</a></li>
<li><a href="#org92c2633">Singular value decomposition</a></li>
</ul>
</div>
</div>
<p>
This is a list of topics that may appear on the final exam. Generally
speaking, anything in our <a href="http://mcrovella.github.io/CS132-Geometric-Algorithms/landing-page.html">textbook</a> is fair game, and this list is not
guaranteed to be exhaustive. The primary aim of this list is to
point out a couple topics that will <i>not</i> appear on the final.
</p>
<div id="outline-container-orgbc2d56d" class="outline-2">
<h2 id="orgbc2d56d">Systems of linear equations</h2>
<div class="outline-text-2" id="text-orgbc2d56d">
</div>
<ul class="org-ul">
<li><a id="orga917676"></a>Counting solutions<br />
<ul class="org-ul">
<li><a id="org50fa117"></a>Consistency<br /></li>
<li><a id="org5d024da"></a>Uniqueness<br /></li>
<li><a id="orga6fe937"></a>Infinite solutions<br /></li>
</ul>
</li>
<li><a id="org5b612aa"></a>Coefficient matrices<br /></li>
<li><a id="org20eb574"></a>Augmented matrices<br /></li>
<li><a id="orgcf05730"></a>Finding solutions<br />
<ul class="org-ul">
<li><a id="orgc287ee5"></a>Elimination<br /></li>
<li><a id="org973e468"></a>Back substitution<br /></li>
</ul>
</li>
<li><a id="orga35ba69"></a>Verifying solutions<br /></li>
<li><a id="orgdd77859"></a>Elementary row operations<br />
<ul class="org-ul">
<li><a id="org7dfa532"></a>Replacement<br /></li>
<li><a id="org65d669b"></a>Interchange<br /></li>
<li><a id="orgcbf7208"></a>Scaling<br /></li>
</ul>
</li>
<li><a id="org0cc4a63"></a>Row Equivalence<br /></li>
</ul>
</div>
<div id="outline-container-org272a31e" class="outline-2">
<h2 id="org272a31e">Gaussian elimination</h2>
<div class="outline-text-2" id="text-org272a31e">
</div>
<ul class="org-ul">
<li><a id="org7fdfbb5"></a>Echelon forms<br /></li>
<li><a id="orgf151862"></a>Reduced echelon forms<br /></li>
<li><a id="org63b32fe"></a>Pivot positions and pivot rows<br /></li>
<li><a id="org8d213fb"></a>Performing Gaussian elimination<br />
<ul class="org-ul">
<li><a id="org35784b6"></a>Elimination<br /></li>
<li><a id="orgc1cb36d"></a>Back substitution<br /></li>
</ul>
</li>
<li><a id="orgdb02402"></a>Interpreting echelon forms<br />
<ul class="org-ul">
<li><a id="org3786191"></a>Basic variables<br /></li>
<li><a id="orgf41b344"></a>Free variables<br /></li>
<li><a id="org9dae62d"></a>General-form solutions<br /></li>
</ul>
</li>
<li><a id="orgef4ec92"></a><b>EXCLUDING</b>: FLOP counts<br /></li>
</ul>
</div>
<div id="outline-container-org1e87871" class="outline-2">
<h2 id="org1e87871">Vector equations</h2>
<div class="outline-text-2" id="text-org1e87871">
</div>
<ul class="org-ul">
<li><a id="org000b839"></a>(Column) vectors<br /></li>
<li><a id="orgc3b1a9f"></a>Vector operations<br />
<ul class="org-ul">
<li><a id="orgca6678b"></a>Addition<br />
<ul class="org-ul">
<li><a id="org344fdb9"></a>Parallelogram rule<br /></li>
<li><a id="org0fa2186"></a>Tip-to-tail rule<br /></li>
</ul>
</li>
<li><a id="org5fb79cb"></a>Multiplication<br /></li>
<li><a id="org1462bd4"></a>Scaling<br /></li>
<li><a id="org8175932"></a>Algebraic properties<br /></li>
</ul>
</li>
<li><a id="orgc99ad2d"></a>Linear combinations<br /></li>
<li><a id="org79359ab"></a>Vector equations<br />
<ul class="org-ul">
<li><a id="org4369882"></a>Equivalence with systems of linear equations<br /></li>
</ul>
</li>
<li><a id="orga996fda"></a>Span<br /></li>
</ul>
</div>
<div id="outline-container-orgb417ad5" class="outline-2">
<h2 id="orgb417ad5">Ax = b</h2>
<div class="outline-text-2" id="text-orgb417ad5">
</div>
<ul class="org-ul">
<li><a id="org28742b0"></a>Matrix-vector multiplication<br />
<ul class="org-ul">
<li><a id="orge15935d"></a>Algebraic properties<br />
<ul class="org-ul">
<li><a id="orga54e1cc"></a>Additivity<br /></li>
<li><a id="org295893e"></a>Homogeneity<br /></li>
</ul>
</li>
</ul>
</li>
<li><a id="orga4c3d42"></a>The matrix equation<br />
<ul class="org-ul">
<li><a id="org16842ae"></a>Equivalence with systems of linear equations<br /></li>
</ul>
</li>
<li><a id="orgf6a7c0f"></a>Inner products<br /></li>
<li><a id="org700305a"></a>The identity matrix<br /></li>
</ul>
</div>
<div id="outline-container-orgb050f7e" class="outline-2">
<h2 id="orgb050f7e">Linear independence</h2>
<div class="outline-text-2" id="text-orgb050f7e">
</div>
<ul class="org-ul">
<li><a id="orge99b660"></a>Linear dependence<br /></li>
<li><a id="org7293f13"></a>Spanning sets<br /></li>
<li><a id="org6415e97"></a><b>EXCLUDING</b>: Network flow<br /></li>
</ul>
</div>
<div id="outline-container-org8b182ab" class="outline-2">
<h2 id="org8b182ab">Linear transforms</h2>
<div class="outline-text-2" id="text-org8b182ab">
</div>
<ul class="org-ul">
<li><a id="org1b1bb70"></a>Definitions<br />
<ul class="org-ul">
<li><a id="orga5319d4"></a>Domain<br /></li>
<li><a id="org26dabed"></a>Codomain<br /></li>
<li><a id="org09cf3ba"></a>Image<br /></li>
<li><a id="orga98bda6"></a>Range<br /></li>
</ul>
</li>
<li><a id="org5e10903"></a>Linearity<br /></li>
<li><a id="org0e26b34"></a>Algebraic properties (of Linear Transformations)<br /></li>
<li><a id="orge4ab3c5"></a>Simple linear transformations<br />
<ul class="org-ul">
<li><a id="org4536a58"></a>Shearing<br /></li>
<li><a id="org32e79ef"></a>Contraction<br /></li>
<li><a id="orgbf36fa8"></a>Dilation<br /></li>
<li><a id="org1e11fa1"></a>Rotation<br /></li>
<li><a id="org36d43f9"></a>Reflection<br /></li>
<li><a id="org1ba3499"></a>Projection<br /></li>
</ul>
</li>
<li><a id="orgeabd045"></a>Simple non-linear transformations<br />
<ul class="org-ul">
<li><a id="org3dcb3f6"></a>Translation<br /></li>
</ul>
</li>
<li><a id="orgc0e94df"></a>Non-geometric word problems<br />
<ul class="org-ul">
<li><a id="org1b8aaf6"></a>Manufacturing example<br /></li>
</ul>
</li>
</ul>
</div>
<div id="outline-container-orge59fbae" class="outline-2">
<h2 id="orge59fbae">Matrix of a transformation</h2>
<div class="outline-text-2" id="text-orge59fbae">
</div>
<ul class="org-ul">
<li><a id="org1c30ea8"></a>Finding the matrix implementing a linear transformation<br /></li>
<li><a id="orgf6db60a"></a>Kinds of linear transformations<br />
<ul class="org-ul">
<li><a id="org97dacb0"></a>One-to-one<br /></li>
<li><a id="orgd38327b"></a>Onto<br /></li>
</ul>
</li>
<li><a id="org0400650"></a>2 × 2 Determinants<br />
<ul class="org-ul">
<li><a id="org32bcac9"></a><b>EXCLUDING</b>: Relationship with area<br /></li>
</ul>
</li>
</ul>
</div>
<div id="outline-container-orgb61fbed" class="outline-2">
<h2 id="orgb61fbed">Matrix algebra</h2>
<div class="outline-text-2" id="text-orgb61fbed">
</div>
<ul class="org-ul">
<li><a id="org5b226f1"></a>Matrix Operations<br />
<ul class="org-ul">
<li><a id="org80919ac"></a>Addition<br /></li>
<li><a id="orgf4ada31"></a>Multiplication<br />
<ul class="org-ul">
<li><a id="orgb0f6172"></a>Relation to composition<br /></li>
</ul>
</li>
<li><a id="org1e39800"></a>Scaling<br /></li>
<li><a id="org23b2700"></a>Algebraic properties<br />
<ul class="org-ul">
<li><a id="org0b4c5b2"></a>Matrix multiplication is not commutative<br /></li>
</ul>
</li>
<li><a id="org4558772"></a>Powers<br /></li>
<li><a id="org2c6278d"></a>Transposition<br /></li>
</ul>
</li>
<li><a id="org7001e5f"></a>Inner product<br /></li>
<li><a id="org4a13ac5"></a><b>EXCLUDING</b>: Computational viewpoint<br /></li>
</ul>
</div>
<div id="outline-container-orgfaed484" class="outline-2">
<h2 id="orgfaed484">Matrix inverses</h2>
<div class="outline-text-2" id="text-orgfaed484">
</div>
<ul class="org-ul">
<li><a id="org34150ff"></a>Invertibility<br /></li>
<li><a id="org815c53e"></a>Singular vs. non-singular matrices<br /></li>
<li><a id="org6bce1df"></a>Determinants<br /></li>
<li><a id="orgf606b93"></a>Equivalence to solving systems of linear equations<br /></li>
<li><a id="org9df6cb6"></a><b>EXCLUDING</b>:<br />
<ul class="org-ul">
<li><a id="org527a998"></a>Computational view<br /></li>
<li><a id="orgf1b2061"></a>Ill-conditioned matrices<br /></li>
</ul>
</li>
</ul>
</div>
<div id="outline-container-org2d760e2" class="outline-2">
<h2 id="org2d760e2">The invertible matrix theorem</h2>
</div>
<div id="outline-container-orgfa98bed" class="outline-2">
<h2 id="orgfa98bed">Markov chains</h2>
<div class="outline-text-2" id="text-orgfa98bed">
</div>
<ul class="org-ul">
<li><a id="org949bb26"></a>Linear dynamical systems/linear difference equations<br /></li>
<li><a id="org9838e56"></a>Probability vectors<br /></li>
<li><a id="org2a0a42d"></a>Stochastic matrices<br /></li>
<li><a id="org98c5c50"></a>Steady-states<br /></li>
<li><a id="orga7de28a"></a>Convergence to steady states<br /></li>
</ul>
</div>
<div id="outline-container-org9caaf71" class="outline-2">
<h2 id="org9caaf71">Matrix factorization</h2>
<div class="outline-text-2" id="text-org9caaf71">
</div>
<ul class="org-ul">
<li><a id="orgba39fd1"></a>LU factorization<br /></li>
<li><a id="org471319c"></a>Elementary matrices<br /></li>
<li><a id="orgb1093ff"></a><b>EXCLUDING</b>:<br />
<ul class="org-ul">
<li><a id="org46925b6"></a>Forward substitution<br /></li>
<li><a id="org77024f1"></a>FLOP counts<br /></li>
<li><a id="org0d2ff93"></a>Pivoting<br /></li>
</ul>
</li>
</ul>
</div>
<div id="outline-container-org86e368c" class="outline-2">
<h2 id="org86e368c">Computer graphics</h2>
<div class="outline-text-2" id="text-org86e368c">
</div>
<ul class="org-ul">
<li><a id="orgd86a151"></a>Homogeneous coordinates<br /></li>
<li><a id="org41a8a26"></a>Rotation matrices<br /></li>
<li><a id="org9ab1548"></a>Perspective projections<br /></li>
<li><a id="org2c462eb"></a>Composing transformation<br /></li>
<li><a id="org4e578cc"></a><b>EXCLUDING</b>: programming aspects of graphics<br /></li>
</ul>
</div>
<div id="outline-container-org6c3bd74" class="outline-2">
<h2 id="org6c3bd74">Subspaces</h2>
<div class="outline-text-2" id="text-org6c3bd74">
</div>
<ul class="org-ul">
<li><a id="orga04acef"></a>Equivalence with spans<br /></li>
<li><a id="org374fb33"></a>Column space<br /></li>
<li><a id="orgaf42f77"></a>Null space<br /></li>
<li><a id="orgc1326df"></a>Basis<br />
<ul class="org-ul">
<li><a id="org61da12c"></a>of column space<br /></li>
<li><a id="org9860152"></a>of null space<br /></li>
</ul>
</li>
</ul>
</div>
<div id="outline-container-org5b8c2dc" class="outline-2">
<h2 id="org5b8c2dc">Dimension and rank</h2>
<div class="outline-text-2" id="text-org5b8c2dc">
</div>
<ul class="org-ul">
<li><a id="orgb08a0cb"></a>Coordinate systems<br /></li>
<li><a id="org02b2837"></a>Dimension of a subspace<br />
<ul class="org-ul">
<li><a id="org8a29706"></a>of column space<br /></li>
<li><a id="orga251d3c"></a>of null space<br /></li>
</ul>
</li>
<li><a id="org2612f86"></a>Rank<br /></li>
<li><a id="org902e423"></a>Rank-nullity theorem<br /></li>
<li><a id="org6f7b0ff"></a><b>EXCLUDING</b>: Isomorphism<br /></li>
</ul>
</div>
<div id="outline-container-orgc4ae8b8" class="outline-2">
<h2 id="orgc4ae8b8">Eigenvectors and eigenvalues</h2>
<div class="outline-text-2" id="text-orgc4ae8b8">
</div>
<ul class="org-ul">
<li><a id="orgdc45128"></a>Checking eigenvectors/values<br /></li>
<li><a id="org795af69"></a>Finding eigenvectors<br /></li>
<li><a id="org2060cad"></a>Eigenspace<br /></li>
<li><a id="org8961587"></a>Eigenvalues of triangular matrices<br /></li>
<li><a id="org50f7580"></a>Invertibility and eigenvalues<br /></li>
<li><a id="org3cef56a"></a>Eigenvalues solve difference equations<br /></li>
</ul>
</div>
<div id="outline-container-org3924a0f" class="outline-2">
<h2 id="org3924a0f">The characteristic equation</h2>
<div class="outline-text-2" id="text-org3924a0f">
</div>
<ul class="org-ul">
<li><a id="org4f6f3ab"></a>Determinants<br /></li>
<li><a id="orgcc0537d"></a>Finding the characteristic polynomial<br /></li>
<li><a id="org6170529"></a>Finding eigenvalues<br /></li>
<li><a id="orge118db5"></a>Similarity<br /></li>
<li><a id="orgc904328"></a>Complete solution for a Markov chain<br /></li>
</ul>
</div>
<div id="outline-container-org30d6a09" class="outline-2">
<h2 id="org30d6a09">Diagonalizability</h2>
<div class="outline-text-2" id="text-org30d6a09">
</div>
<ul class="org-ul">
<li><a id="org5f6cec1"></a>Diagonal matrices<br /></li>
<li><a id="org2afd2ee"></a>Similar Matrices<br /></li>
<li><a id="org04e44c0"></a>Diagonalizability<br /></li>
<li><a id="org36d5ba0"></a>Eigenbasis<br /></li>
<li><a id="orga4d7ba3"></a>Exposing the behavior of dynamical systems<br /></li>
</ul>
</div>
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<h2 id="orgd2109bd">PageRank</h2>
<div class="outline-text-2" id="text-orgd2109bd">
</div>
<ul class="org-ul">
<li><a id="org17c20ce"></a>Random walks<br /></li>
<li><a id="org595c660"></a><b>EXCLUDING</b>:<br />
<ul class="org-ul">
<li><a id="orge275088"></a>The history<br /></li>
<li><a id="org727e1c2"></a>Reflecting vs. absorbing boundaries<br /></li>
<li><a id="org697464f"></a>The power method<br /></li>
</ul>
</li>
</ul>
</div>
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<h2 id="orgad264ae">Analytic geometry</h2>
<div class="outline-text-2" id="text-orgad264ae">
</div>
<ul class="org-ul">
<li><a id="orgc79a32a"></a>Inner product/dot product<br /></li>
<li><a id="org18ef07e"></a>Norm<br /></li>
<li><a id="org1231d4c"></a>Distance<br /></li>
<li><a id="orgc4f78fa"></a>Angle<br /></li>
<li><a id="org1713f9e"></a>Orthogonality<br /></li>
<li><a id="orgcf4b892"></a>Cosine similarity<br /></li>
</ul>
</div>
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<h2 id="orgcfde3b3">Orthogonal projection</h2>
<div class="outline-text-2" id="text-orgcfde3b3">
</div>
<ul class="org-ul">
<li><a id="org7382548"></a>Orthogonal basis<br /></li>
<li><a id="orgc2d48a6"></a>Orthogonal projection<br /></li>
<li><a id="orgce764e9"></a>Projections and coordinates<br /></li>
<li><a id="org018efe9"></a>Orthonormal sets<br /></li>
<li><a id="org099bf6d"></a>Orthogonal matrices<br /></li>
</ul>
</div>
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<h2 id="org6a2191c">Least squares</h2>
<div class="outline-text-2" id="text-org6a2191c">
</div>
<ul class="org-ul">
<li><a id="orge13afc6"></a>Finding least squares<br /></li>
<li><a id="org89c0fc3"></a>Normal equations<br /></li>
<li><a id="org1eaadcf"></a>Projecting onto a basis<br /></li>
</ul>
</div>
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<h2 id="orgd5c348f">Linear models</h2>
<div class="outline-text-2" id="text-orgd5c348f">
</div>
<ul class="org-ul">
<li><a id="orgc6d1a16"></a>Best fit line/quadratic<br /></li>
<li><a id="orgedb8e21"></a>Multiple regression<br /></li>
<li><a id="org1caeb1c"></a>Design matrices<br /></li>
</ul>
</div>
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<h2 id="org43b8af7">Symmetric matrices</h2>
<div class="outline-text-2" id="text-org43b8af7">
</div>
<ul class="org-ul">
<li><a id="org9756006"></a>Orthogonal diagonalization<br /></li>
<li><a id="org495fd76"></a>Quadratic forms<br /></li>
<li><a id="org615b073"></a>Definiteness<br /></li>
<li><a id="org1cb80af"></a>Constrained optimization<br /></li>
</ul>
</div>
<div id="outline-container-org92c2633" class="outline-2">
<h2 id="org92c2633">Singular value decomposition</h2>
<div class="outline-text-2" id="text-org92c2633">
</div>
<ul class="org-ul">
<li><a id="orged218cf"></a>\(\|A\mathbf x\|^2\) as a quadratic form<br /></li>
<li><a id="orgb2c3d0d"></a>Singular values<br /></li>
<li><a id="org6ceff43"></a>Determining the SVD of a matrix<br /></li>
<li><a id="orgb772699"></a>Reduced SVD and the psuedoinverse<br /></li>
<li><a id="orgb8e0364"></a><b>EXCLUDING</b><br />
<ul class="org-ul">
<li><a id="org63e6a7e"></a>Applications of the SVD<br /></li>
</ul>
</li>
</ul>
</div>
</div>
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