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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 1.00
</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 <span class="obeylines-h"><a 
href="http://linear.ups.edu" class="url" ><span 
class="cmtt-12">http://linear.ups.edu</span></a></span>.<br class="newline" />
                                                                          

                                                                          
</p><!--l. 239--><p class="indent">
                                                                          

                                                                          
</p>
   <div class="tableofcontents">
   <span class="likechapterToc"><a 
href="fcla-xml-1.00li1.xml#x2-1000" id="QQ2-2-1">Front Matter</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li2.xml#x3-2000" id="QQ2-3-2">Author Biography</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li3.xml#x4-3000" id="QQ2-4-3">Edition</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li4.xml#x5-4000" id="QQ2-5-4">Publisher</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li5.xml#x6-5000" id="QQ2-6-5">Dedication</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li6.xml#x7-6000" id="QQ2-7-6">Contributors</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li7.xml#x8-7000" id="QQ2-8-7">Definitions</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li8.xml#x9-8000" id="QQ2-9-8">Theorems</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li9.xml#x10-9000" id="QQ2-10-9">Notation</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li10.xml#x11-10000" id="QQ2-11-10">Examples</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li11.xml#x12-11000" id="QQ2-12-11">Preface</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li12.xml#x13-17000" id="QQ2-13-17">Acknowledgements</a></span>
<br />   <span class="likepartToc"><a 
href="fcla-xml-1.00li13.xml#x14-18000" id="QQ2-14-18">Part C Core</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li14.xml#x15-19000" id="QQ2-15-19">Chapter SLE Systems of Linear Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li15.xml#x16-20000" id="QQ2-16-20">Section WILA What is Linear Algebra?</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li16.xml#x17-26000" id="QQ2-17-26">Section SSLE Solving Systems of Linear Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li17.xml#x18-33000" id="QQ2-18-33">Section RREF Reduced Row-Echelon Form</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li18.xml#x19-40000" id="QQ2-19-40">Section TSS Types of Solution Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li19.xml#x20-46000" id="QQ2-20-46">Section HSE Homogeneous Systems of Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li20.xml#x21-52000" id="QQ2-21-52">Section NM Nonsingular Matrices</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li21.xml#x22-58000" id="QQ2-22-58">Chapter V Vectors</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li22.xml#x23-59000" id="QQ2-23-59">Section VO Vector Operations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li23.xml#x24-65000" id="QQ2-24-65">Section LC Linear Combinations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li24.xml#x25-73000" id="QQ2-25-73">Section SS Spanning Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li25.xml#x26-79000" id="QQ2-26-79">Section LI Linear Independence</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li26.xml#x27-86000" id="QQ2-27-86">Section LDS Linear Dependence and Spans</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li27.xml#x28-92000" id="QQ2-28-92">Section O Orthogonality</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li28.xml#x29-100000" id="QQ2-29-100">Chapter M Matrices</a></span>
                                                                          

                                                                          
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li29.xml#x30-101000" id="QQ2-30-101">Section MO Matrix Operations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li30.xml#x31-109000" id="QQ2-31-109">Section MM Matrix Multiplication</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li31.xml#x32-117000" id="QQ2-32-117">Section MISLE Matrix Inverses and Systems of Linear Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li32.xml#x33-124000" id="QQ2-33-124">Section MINM Matrix Inverses and Nonsingular Matrices</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li33.xml#x34-130000" id="QQ2-34-130">Section CRS Column and Row Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li34.xml#x35-138000" id="QQ2-35-138">Section FS Four Subsets</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li35.xml#x36-146000" id="QQ2-36-146">Chapter VS Vector Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li36.xml#x37-147000" id="QQ2-37-147">Section VS Vector Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li37.xml#x38-154000" id="QQ2-38-154">Section S Subspaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li38.xml#x39-161000" id="QQ2-39-161">Section LISS Linear Independence and Spanning Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li39.xml#x40-168000" id="QQ2-40-168">Section B Bases</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li40.xml#x41-176000" id="QQ2-41-176">Section D Dimension</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li41.xml#x42-184000" id="QQ2-42-184">Section PD Properties of Dimension</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li42.xml#x43-191000" id="QQ2-43-191">Chapter D Determinants</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li43.xml#x44-192000" id="QQ2-44-192">Section DM Determinant of a Matrix</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li44.xml#x45-199000" id="QQ2-45-199">Section PDM Properties of Determinants of Matrices</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li45.xml#x46-206000" id="QQ2-46-206">Chapter E Eigenvalues</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li46.xml#x47-207000" id="QQ2-47-207">Section EE Eigenvalues and Eigenvectors</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li47.xml#x48-216000" id="QQ2-48-216">Section PEE Properties of Eigenvalues and Eigenvectors</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li48.xml#x49-222000" id="QQ2-49-222">Section SD Similarity and Diagonalization</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li49.xml#x50-230000" id="QQ2-50-230">Chapter LT Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li50.xml#x51-231000" id="QQ2-51-231">Section LT Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li51.xml#x52-240000" id="QQ2-52-240">Section ILT Injective Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li52.xml#x53-249000" id="QQ2-53-249">Section SLT Surjective Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li53.xml#x54-258000" id="QQ2-54-258">Section IVLT Invertible Linear Transformations</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li54.xml#x55-267000" id="QQ2-55-267">Chapter R Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li55.xml#x56-268000" id="QQ2-56-268">Section VR Vector Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li56.xml#x57-275000" id="QQ2-57-275">Section MR Matrix Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li57.xml#x58-282000" id="QQ2-58-282">Section CB Change of Basis</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li58.xml#x59-290000" id="QQ2-59-290">Appendix CN Computation Notes</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li59.xml#x60-291000" id="QQ2-60-291">Section MMA Mathematica</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li60.xml#x61-302000" id="QQ2-61-302">Section TI86 Texas Instruments 86</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li61.xml#x62-307000" id="QQ2-62-307">Section TI83 Texas Instruments 83</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li62.xml#x63-311000" id="QQ2-63-311">Appendix P Preliminaries</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li63.xml#x64-312000" id="QQ2-64-312">Section CNO Complex Number Operations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li64.xml#x65-316000" id="QQ2-65-316">Section SET Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li65.xml#x66-319000" id="QQ2-66-319">Section PT Proof Techniques</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li66.xml#x67-337000" id="QQ2-67-337">Appendix A Archetypes</a></span>
                                                                          

                                                                          
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li67.xml#x68-338000" id="QQ2-68-338">Archetype A </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li68.xml#x69-340000" id="QQ2-69-340">Archetype B </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li69.xml#x70-342000" id="QQ2-70-342">Archetype C </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li70.xml#x71-344000" id="QQ2-71-344">Archetype D </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li71.xml#x72-346000" id="QQ2-72-346">Archetype E </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li72.xml#x73-348000" id="QQ2-73-348">Archetype F </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li73.xml#x74-350000" id="QQ2-74-350">Archetype G </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li74.xml#x75-352000" id="QQ2-75-352">Archetype H </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li75.xml#x76-354000" id="QQ2-76-354">Archetype I </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li76.xml#x77-356000" id="QQ2-77-356">Archetype J </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li77.xml#x78-358000" id="QQ2-78-358">Archetype K </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li78.xml#x79-360000" id="QQ2-79-360">Archetype L </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li79.xml#x80-362000" id="QQ2-80-362">Archetype M </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li80.xml#x81-364000" id="QQ2-81-364">Archetype N </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li81.xml#x82-366000" id="QQ2-82-366">Archetype O </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li82.xml#x83-368000" id="QQ2-83-368">Archetype P </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li83.xml#x84-370000" id="QQ2-84-370">Archetype Q </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li84.xml#x85-372000" id="QQ2-85-372">Archetype R </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li85.xml#x86-374000" id="QQ2-86-374">Archetype S </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li86.xml#x87-376000" id="QQ2-87-376">Archetype T </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li87.xml#x88-378000" id="QQ2-88-378">Archetype U </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li88.xml#x89-380000" id="QQ2-89-380">Archetype V </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li89.xml#x90-382000" id="QQ2-90-382">Archetype W </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.00li90.xml#x91-384000" id="QQ2-91-384">Archetype X </a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li91.xml#x92-386000" id="QQ2-92-386">Appendix GFDL GNU Free Documentation License</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-387">1. APPLICABILITY AND DEFINITIONS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-388">2. VERBATIM COPYING</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-389">3. COPYING IN QUANTITY</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-390">4. MODIFICATIONS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-391">5. COMBINING DOCUMENTS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-392">6. COLLECTIONS OF DOCUMENTS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-393">7. AGGREGATION WITH INDEPENDENT WORKS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-394">8. TRANSLATION</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-395">9. TERMINATION</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-396">10. FUTURE REVISIONS OF THIS LICENSE</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.00li91.xml#Q1-92-397">ADDENDUM: How to use this License for your documents</a></span>
<br />   <span class="likepartToc"><a 
href="fcla-xml-1.00li92.xml#x93-387000" id="QQ2-93-398">Part T Topics</a></span>
<br />   <span class="likepartToc"><a 
href="fcla-xml-1.00li93.xml#x94-388000" id="QQ2-94-399">Part A Applications</a></span>
                                                                          

                                                                          
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.00li94.xml#x95-389000" id="QQ2-95-400">Index</a></span>
   </div>
                                                                          

                                                                          
                                                                          

                                                                          

                                                                          

                                                                          

                                                                          

                                                                          
                                                                          

                                                                          

                                                                          

                                                                          

                                                                          

                                                                          
                                                                          

                                                                          

                                                                          

                                                                          
                                                                          

                                                                          

                                                                          

                                                                          

                                                                          

                                                                          
                                                                          

                                                                          
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