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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 1.01
</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-1.01li1.xml#x2-1000" id="QQ2-2-1">Front Matter</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li2.xml#x3-2000" id="QQ2-3-2">Author Biography</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li3.xml#x4-3000" id="QQ2-4-3">Edition</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li4.xml#x5-4000" id="QQ2-5-4">Publisher</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li5.xml#x6-5000" id="QQ2-6-5">Dedication</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li6.xml#x7-6000" id="QQ2-7-6">Contributors</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li7.xml#x8-7000" id="QQ2-8-7">Definitions</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li8.xml#x9-8000" id="QQ2-9-8">Theorems</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li9.xml#x10-9000" id="QQ2-10-9">Notation</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li10.xml#x11-10000" id="QQ2-11-10">Examples</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li11.xml#x12-11000" id="QQ2-12-11">Preface</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li12.xml#x13-17000" id="QQ2-13-17">Acknowledgements</a></span>
<br />   <span class="likepartToc"><a 
href="fcla-xml-1.01li13.xml#x14-18000" id="QQ2-14-18">Part C Core</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li14.xml#x15-19000" id="QQ2-15-19">Chapter SLE Systems of Linear Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li15.xml#x16-20000" id="QQ2-16-20">Section WILA What is Linear Algebra?</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li16.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-1.01li17.xml#x18-33000" id="QQ2-18-33">Section RREF Reduced Row-Echelon Form</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li18.xml#x19-40000" id="QQ2-19-40">Section TSS Types of Solution Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li19.xml#x20-46000" id="QQ2-20-46">Section HSE Homogeneous Systems of Equations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li20.xml#x21-52000" id="QQ2-21-52">Section NM Nonsingular Matrices</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li21.xml#x22-58000" id="QQ2-22-58">Chapter V Vectors</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li22.xml#x23-59000" id="QQ2-23-59">Section VO Vector Operations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li23.xml#x24-65000" id="QQ2-24-65">Section LC Linear Combinations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li24.xml#x25-73000" id="QQ2-25-73">Section SS Spanning Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li25.xml#x26-79000" id="QQ2-26-79">Section LI Linear Independence</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li26.xml#x27-86000" id="QQ2-27-86">Section LDS Linear Dependence and Spans</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li27.xml#x28-92000" id="QQ2-28-92">Section O Orthogonality</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li28.xml#x29-100000" id="QQ2-29-100">Chapter M Matrices</a></span>
                                                                          

                                                                          
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li29.xml#x30-101000" id="QQ2-30-101">Section MO Matrix Operations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li30.xml#x31-109000" id="QQ2-31-109">Section MM Matrix Multiplication</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li31.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-1.01li32.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-1.01li33.xml#x34-130000" id="QQ2-34-130">Section CRS Column and Row Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li34.xml#x35-138000" id="QQ2-35-138">Section FS Four Subsets</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li35.xml#x36-146000" id="QQ2-36-146">Chapter VS Vector Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li36.xml#x37-147000" id="QQ2-37-147">Section VS Vector Spaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li37.xml#x38-154000" id="QQ2-38-154">Section S Subspaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li38.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-1.01li39.xml#x40-168000" id="QQ2-40-168">Section B Bases</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li40.xml#x41-176000" id="QQ2-41-176">Section D Dimension</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li41.xml#x42-184000" id="QQ2-42-184">Section PD Properties of Dimension</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li42.xml#x43-192000" id="QQ2-43-192">Chapter D Determinants</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li43.xml#x44-193000" id="QQ2-44-193">Section DM Determinant of a Matrix</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li44.xml#x45-200000" id="QQ2-45-200">Section PDM Properties of Determinants of Matrices</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li45.xml#x46-207000" id="QQ2-46-207">Chapter E Eigenvalues</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li46.xml#x47-208000" id="QQ2-47-208">Section EE Eigenvalues and Eigenvectors</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li47.xml#x48-217000" id="QQ2-48-217">Section PEE Properties of Eigenvalues and Eigenvectors</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li48.xml#x49-223000" id="QQ2-49-223">Section SD Similarity and Diagonalization</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li49.xml#x50-231000" id="QQ2-50-231">Chapter LT Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li50.xml#x51-232000" id="QQ2-51-232">Section LT Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li51.xml#x52-241000" id="QQ2-52-241">Section ILT Injective Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li52.xml#x53-250000" id="QQ2-53-250">Section SLT Surjective Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li53.xml#x54-259000" id="QQ2-54-259">Section IVLT Invertible Linear Transformations</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li54.xml#x55-268000" id="QQ2-55-268">Chapter R Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li55.xml#x56-269000" id="QQ2-56-269">Section VR Vector Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li56.xml#x57-276000" id="QQ2-57-276">Section MR Matrix Representations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li57.xml#x58-283000" id="QQ2-58-283">Section CB Change of Basis</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li58.xml#x59-291000" id="QQ2-59-291">Section NLT Nilpotent Linear Transformations</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li59.xml#x60-295000" id="QQ2-60-295">Section IS Invariant Subspaces</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li60.xml#x61-299000" id="QQ2-61-299">Section JCF Jordan Canonical Form</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li61.xml#x62-304000" id="QQ2-62-304">Appendix CN Computation Notes</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li62.xml#x63-305000" id="QQ2-63-305">Section MMA Mathematica</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li63.xml#x64-316000" id="QQ2-64-316">Section TI86 Texas Instruments 86</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li64.xml#x65-321000" id="QQ2-65-321">Section TI83 Texas Instruments 83</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li65.xml#x66-325000" id="QQ2-66-325">Appendix P Preliminaries</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li66.xml#x67-326000" id="QQ2-67-326">Section CNO Complex Number Operations</a></span>
                                                                          

                                                                          
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li67.xml#x68-330000" id="QQ2-68-330">Section SET Sets</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li68.xml#x69-333000" id="QQ2-69-333">Section PT Proof Techniques</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li69.xml#x70-351000" id="QQ2-70-351">Appendix A Archetypes</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li70.xml#x71-352000" id="QQ2-71-352">Archetype A </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li71.xml#x72-354000" id="QQ2-72-354">Archetype B </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li72.xml#x73-356000" id="QQ2-73-356">Archetype C </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li73.xml#x74-358000" id="QQ2-74-358">Archetype D </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li74.xml#x75-360000" id="QQ2-75-360">Archetype E </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li75.xml#x76-362000" id="QQ2-76-362">Archetype F </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li76.xml#x77-364000" id="QQ2-77-364">Archetype G </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li77.xml#x78-366000" id="QQ2-78-366">Archetype H </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li78.xml#x79-368000" id="QQ2-79-368">Archetype I </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li79.xml#x80-370000" id="QQ2-80-370">Archetype J </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li80.xml#x81-372000" id="QQ2-81-372">Archetype K </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li81.xml#x82-374000" id="QQ2-82-374">Archetype L </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li82.xml#x83-376000" id="QQ2-83-376">Archetype M </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li83.xml#x84-378000" id="QQ2-84-378">Archetype N </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li84.xml#x85-380000" id="QQ2-85-380">Archetype O </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li85.xml#x86-382000" id="QQ2-86-382">Archetype P </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li86.xml#x87-384000" id="QQ2-87-384">Archetype Q </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li87.xml#x88-386000" id="QQ2-88-386">Archetype R </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li88.xml#x89-388000" id="QQ2-89-388">Archetype S </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li89.xml#x90-390000" id="QQ2-90-390">Archetype T </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li90.xml#x91-392000" id="QQ2-91-392">Archetype U </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li91.xml#x92-394000" id="QQ2-92-394">Archetype V </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li92.xml#x93-396000" id="QQ2-93-396">Archetype W </a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li93.xml#x94-398000" id="QQ2-94-398">Archetype X </a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li94.xml#x95-400000" id="QQ2-95-400">Appendix GFDL GNU Free Documentation License</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-401">1. APPLICABILITY AND DEFINITIONS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-402">2. VERBATIM COPYING</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-403">3. COPYING IN QUANTITY</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-404">4. MODIFICATIONS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-405">5. COMBINING DOCUMENTS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-406">6. COLLECTIONS OF DOCUMENTS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-407">7. AGGREGATION WITH INDEPENDENT WORKS</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-408">8. TRANSLATION</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-409">9. TERMINATION</a></span>
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-410">10. FUTURE REVISIONS OF THIS LICENSE</a></span>
                                                                          

                                                                          
<br />   &#x00A0;<span class="sectionToc"> <a 
href="fcla-xml-1.01li94.xml#Q1-95-411">ADDENDUM: How to use this License for your documents</a></span>
<br />   <span class="likepartToc"><a 
href="fcla-xml-1.01li95.xml#x96-401000" id="QQ2-96-412">Part T Topics</a></span>
<br />   &#x00A0;<span class="likesectionToc"><a 
href="fcla-xml-1.01li96.xml#x97-402000" id="QQ2-97-413">Section F Fields</a></span>
<br />   <span class="likepartToc"><a 
href="fcla-xml-1.01li97.xml#x98-408000" id="QQ2-98-419">Part A Applications</a></span>
<br />   <span class="likechapterToc"><a 
href="fcla-xml-1.01li98.xml#x99-409000" id="QQ2-99-420">Index</a></span>
   </div>
                                                                          

                                                                          
                                                                          

                                                                          

                                                                          

                                                                          

                                                                          

                                                                          
                                                                          

                                                                          

                                                                          

                                                                          
                                                                          

                                                                          

                                                                          

                                                                          
                                                                          

                                                                          

                                                                          

                                                                          

                                                                          

                                                                          

                                                                          

                                                                          
                                                                          

                                                                          
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