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<!--l. 10--><p class="noindent" >
</p><!--l. 12--><p class="noindent" ><span 
class="cmr-17x-x-143">A First Course in Linear Algebra</span>
</p><!--l. 16--><p class="noindent" ><span 
class="cmr-12x-x-120">by</span><br />
<span 
class="cmr-12x-x-120">Robert A.</span><span 
class="cmr-12x-x-120">&#x00A0;Beezer</span><br />
<span 
class="cmr-12x-x-120">Department of Mathematics and Computer Science</span><br />
<span 
class="cmr-12x-x-120">University of Puget Sound</span>
</p><!--l. 20--><p class="noindent" >Version 1.35
</p>
</div>
<!--l. 10--><p class="noindent" ><span 
class="cmsy-10x-x-120">&#x00A9;</span>&#x00A0;2004 by Robert A. Beezer.<br class="newline" />
Permission is granted to copy, distribute and/or modify this document under the
terms of the GNU Free Documentation License, Version 1.2 or any later version
published by the Free Software Foundation; with the Invariant Sections being
&#x201C;Preface&#x201D;, no Front-Cover Texts, and no Back-Cover Texts. A copy of the
license is included in the appendix entitled &#x201C;GNU Free Documentation
License&#x201D;.<br class="newline" />
The most recent version of this work can always be found at <a 
href="http://linear.ups.edu" class="url" ><span 
class="cmtt-12">http://linear.ups.edu</span></a>.<br class="newline" />
                                                                          

                                                                          
</p><!--l. 245--><p class="indent" >
                                                                          

                                                                          
</p>
   <div class="tableofcontents">
   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li1.xml#x2-1000" id="QQ2-2-1">Front Matter</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li2.xml#x3-2000" id="QQ2-3-2">Author Biography</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li3.xml#x4-3000" id="QQ2-4-3">Edition</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li4.xml#x5-4000" id="QQ2-5-4">Publisher</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li5.xml#x6-5000" id="QQ2-6-5">Dedication</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li6.xml#x7-6000" id="QQ2-7-6">Contributors</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li7.xml#x8-7000" id="QQ2-8-7">Definitions</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li8.xml#x9-8000" id="QQ2-9-8">Theorems</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li9.xml#x10-9000" id="QQ2-10-9">Notation</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li10.xml#x11-10000" id="QQ2-11-10">Figures</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li11.xml#x12-11000" id="QQ2-12-11">Examples</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li12.xml#x13-12000" id="QQ2-13-12">Preface</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li13.xml#x14-18000" id="QQ2-14-18">Acknowledgements</a></span>
<br />   <span class="likepartToc" ><a 
href="fcla-xml-1.35li14.xml#x15-19000" id="QQ2-15-19">Part C Core</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li15.xml#x16-20000" id="QQ2-16-20">Chapter SLE Systems of Linear Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li16.xml#x17-21000" id="QQ2-17-21">Section WILA What is Linear Algebra?</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li17.xml#x18-27000" id="QQ2-18-27">Section SSLE Solving Systems of Linear Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li18.xml#x19-34000" id="QQ2-19-34">Section RREF Reduced Row-Echelon Form</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li19.xml#x20-41000" id="QQ2-20-41">Section TSS Types of Solution Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li20.xml#x21-47000" id="QQ2-21-47">Section HSE Homogeneous Systems of Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li21.xml#x22-53000" id="QQ2-22-53">Section NM Nonsingular Matrices</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li22.xml#x23-60000" id="QQ2-23-60">Chapter V Vectors</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li23.xml#x24-61000" id="QQ2-24-61">Section VO Vector Operations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li24.xml#x25-67000" id="QQ2-25-67">Section LC Linear Combinations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li25.xml#x26-74000" id="QQ2-26-74">Section SS Spanning Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li26.xml#x27-80000" id="QQ2-27-80">Section LI Linear Independence</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li27.xml#x28-87000" id="QQ2-28-87">Section LDS Linear Dependence and Spans</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li28.xml#x29-93000" id="QQ2-29-93">Section O Orthogonality</a></span>
                                                                          

                                                                          
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li29.xml#x30-103000" id="QQ2-30-103">Chapter M Matrices</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li30.xml#x31-104000" id="QQ2-31-104">Section MO Matrix Operations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li31.xml#x32-113000" id="QQ2-32-113">Section MM Matrix Multiplication</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li32.xml#x33-122000" id="QQ2-33-122">Section MISLE Matrix Inverses and Systems of Linear Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li33.xml#x34-129000" id="QQ2-34-129">Section MINM Matrix Inverses and Nonsingular Matrices</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li34.xml#x35-135000" id="QQ2-35-135">Section CRS Column and Row Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li35.xml#x36-143000" id="QQ2-36-143">Section FS Four Subsets</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li36.xml#x37-152000" id="QQ2-37-152">Chapter VS Vector Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li37.xml#x38-153000" id="QQ2-38-153">Section VS Vector Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li38.xml#x39-161000" id="QQ2-39-161">Section S Subspaces</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li39.xml#x40-168000" id="QQ2-40-168">Section LISS Linear Independence and Spanning Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li40.xml#x41-175000" id="QQ2-41-175">Section B Bases</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li41.xml#x42-183000" id="QQ2-42-183">Section D Dimension</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li42.xml#x43-191000" id="QQ2-43-191">Section PD Properties of Dimension</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li43.xml#x44-200000" id="QQ2-44-200">Chapter D Determinants</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li44.xml#x45-201000" id="QQ2-45-201">Section DM Determinant of a Matrix</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li45.xml#x46-208000" id="QQ2-46-208">Section PDM Properties of Determinants of Matrices</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li46.xml#x47-216000" id="QQ2-47-216">Chapter E Eigenvalues</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li47.xml#x48-217000" id="QQ2-48-217">Section EE Eigenvalues and Eigenvectors</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li48.xml#x49-226000" id="QQ2-49-226">Section PEE Properties of Eigenvalues and Eigenvectors</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li49.xml#x50-232000" id="QQ2-50-232">Section SD Similarity and Diagonalization</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li50.xml#x51-241000" id="QQ2-51-241">Chapter LT Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li51.xml#x52-242000" id="QQ2-52-242">Section LT Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li52.xml#x53-252000" id="QQ2-53-252">Section ILT Injective Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li53.xml#x54-261000" id="QQ2-54-261">Section SLT Surjective Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li54.xml#x55-270000" id="QQ2-55-270">Section IVLT Invertible Linear Transformations</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li55.xml#x56-280000" id="QQ2-56-280">Chapter R Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li56.xml#x57-281000" id="QQ2-57-281">Section VR Vector Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li57.xml#x58-288000" id="QQ2-58-288">Section MR Matrix Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li58.xml#x59-295000" id="QQ2-59-295">Section CB Change of Basis</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li59.xml#x60-303000" id="QQ2-60-303">Section OD Orthonormal Diagonalization</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li60.xml#x61-308000" id="QQ2-61-308">Section NLT Nilpotent Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li61.xml#x62-312000" id="QQ2-62-312">Section IS Invariant Subspaces</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li62.xml#x63-316000" id="QQ2-63-316">Section JCF Jordan Canonical Form</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li63.xml#x64-321000" id="QQ2-64-321">Appendix CN Computation Notes</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li64.xml#x65-322000" id="QQ2-65-322">Section MMA Mathematica</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li65.xml#x66-333000" id="QQ2-66-333">Section TI86 Texas Instruments 86</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li66.xml#x67-338000" id="QQ2-67-338">Section TI83 Texas Instruments 83</a></span>
                                                                          

                                                                          
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li67.xml#x68-342000" id="QQ2-68-342">Section SAGE SAGE: Open Source Mathematics Software</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li68.xml#x69-348000" id="QQ2-69-348">Appendix P Preliminaries</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li69.xml#x70-349000" id="QQ2-70-349">Section CNO Complex Number Operations</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li70.xml#x71-353000" id="QQ2-71-353">Section SET Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li71.xml#x72-356000" id="QQ2-72-356">Section PT Proof Techniques</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li72.xml#x73-374000" id="QQ2-73-374">Appendix A Archetypes</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li73.xml#x74-375000" id="QQ2-74-375">Archetype A </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li74.xml#x75-377000" id="QQ2-75-377">Archetype B </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li75.xml#x76-379000" id="QQ2-76-379">Archetype C </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li76.xml#x77-381000" id="QQ2-77-381">Archetype D </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li77.xml#x78-383000" id="QQ2-78-383">Archetype E </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li78.xml#x79-385000" id="QQ2-79-385">Archetype F </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li79.xml#x80-387000" id="QQ2-80-387">Archetype G </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li80.xml#x81-389000" id="QQ2-81-389">Archetype H </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li81.xml#x82-391000" id="QQ2-82-391">Archetype I </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li82.xml#x83-393000" id="QQ2-83-393">Archetype J </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li83.xml#x84-395000" id="QQ2-84-395">Archetype K </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li84.xml#x85-397000" id="QQ2-85-397">Archetype L </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li85.xml#x86-399000" id="QQ2-86-399">Archetype M </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li86.xml#x87-401000" id="QQ2-87-401">Archetype N </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li87.xml#x88-403000" id="QQ2-88-403">Archetype O </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li88.xml#x89-405000" id="QQ2-89-405">Archetype P </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li89.xml#x90-407000" id="QQ2-90-407">Archetype Q </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li90.xml#x91-409000" id="QQ2-91-409">Archetype R </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li91.xml#x92-411000" id="QQ2-92-411">Archetype S </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li92.xml#x93-413000" id="QQ2-93-413">Archetype T </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li93.xml#x94-415000" id="QQ2-94-415">Archetype U </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li94.xml#x95-417000" id="QQ2-95-417">Archetype V </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li95.xml#x96-419000" id="QQ2-96-419">Archetype W </a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li96.xml#x97-421000" id="QQ2-97-421">Archetype X </a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li97.xml#x98-423000" id="QQ2-98-423">Appendix GFDL GNU Free Documentation License</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-424">1. APPLICABILITY AND DEFINITIONS</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-425">2. VERBATIM COPYING</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-426">3. COPYING IN QUANTITY</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-427">4. MODIFICATIONS</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-428">5. COMBINING DOCUMENTS</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-429">6. COLLECTIONS OF DOCUMENTS</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-430">7. AGGREGATION WITH INDEPENDENT WORKS</a></span>
                                                                          

                                                                          
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-431">8. TRANSLATION</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-432">9. TERMINATION</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-433">10. FUTURE REVISIONS OF THIS LICENSE</a></span>
<br />   &#x00A0;<span class="sectionToc" > <a 
href="fcla-xml-1.35li97.xml#Q1-98-434">ADDENDUM: How to use this License for your documents</a></span>
<br />   <span class="likepartToc" ><a 
href="fcla-xml-1.35li98.xml#x99-424000" id="QQ2-99-435">Part T Topics</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li99.xml#x100-425000" id="QQ2-100-436">Section F Fields</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li100.xml#x101-431000" id="QQ2-101-442">Section T Trace</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li101.xml#x102-434000" id="QQ2-102-445">Section HP Hadamard Product</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li102.xml#x103-437000" id="QQ2-103-448">Section VM Vandermonde Matrix</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li103.xml#x104-438000" id="QQ2-104-449">Section PSM Positive Semi-definite Matrices</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li104.xml#x105-441000" id="QQ2-105-452">Chapter MD Matrix Decompositions</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li105.xml#x106-442000" id="QQ2-106-453">Section ROD Rank One Decomposition</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li106.xml#x107-443000" id="QQ2-107-454">Section TD Triangular Decomposition</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li107.xml#x108-447000" id="QQ2-108-458">Section SVD Singular Value Decomposition</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li108.xml#x109-450000" id="QQ2-109-461">Section SR Square Roots</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li109.xml#x110-452000" id="QQ2-110-463">Section POD Polar Decomposition</a></span>
<br />   <span class="likepartToc" ><a 
href="fcla-xml-1.35li110.xml#x111-453000" id="QQ2-111-464">Part A Applications</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li111.xml#x112-454000" id="QQ2-112-465">Section CF Curve Fitting</a></span>
<br />   &#x00A0;<span class="likesectionToc" ><a 
href="fcla-xml-1.35li112.xml#x113-457000" id="QQ2-113-468">Section SAS Sharing A Secret</a></span>
<br />   <span class="likechapterToc" ><a 
href="fcla-xml-1.35li113.xml#x114-458000" id="QQ2-114-469">Index</a></span>
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