53Linear-Algebra_opt.zip

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Mathematics-LINEAR ALGEBRA
53Linear-Algebra_opt.zip
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<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/622b80033d2fbb00077d7016/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622b80033d2fbb00077d7016/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">GATE Material for Mat<span class="_ _0"></span>hematics</div><div class="t m0 x1 h3 y2 ff2 fs1 fc0 sc0 ls0 ws0">Lesson 1:</div><div class="t m0 x1 h3 y3 ff2 fs1 fc0 sc0 ls0 ws0">Linear Algebra<span class="ff1">:</span></div><div class="t m0 x1 h3 y4 ff2 fs1 fc0 sc0 ls0 ws0">Introduction to Matrices:</div><div class="t m0 x1 h4 y5 ff1 fs1 fc0 sc0 ls0 ws0">Matrices <span class="_ _1"> </span>and <span class="_ _1"> </span>Determinants <span class="_ _1"> </span>were <span class="_ _1"> </span>discovered <span class="_ _1"> </span>and <span class="_ _1"> </span>developed <span class="_ _1"> </span>in <span class="_ _1"> </span>the <span class="_ _1"> </span>eighteenth <span class="_ _1"> </span>and </div><div class="t m0 x1 h4 y6 ff1 fs1 fc0 sc0 ls0 ws0">nineteenth<span class="_"> </span>centuries.<span class="_"> </span>Initially, <span class="_ _2"></span>their <span class="_ _2"></span>development <span class="_ _3"></span>dealt <span class="_ _2"></span>with <span class="_ _3"></span>transformation <span class="_ _2"></span>of <span class="_ _3"></span>geometric <span class="_ _2"></span>objects </div><div class="t m0 x1 h4 y7 ff1 fs1 fc0 sc0 ls0 ws0">and <span class="_ _4"> </span>solution <span class="_ _4"> </span>of <span class="_ _4"> </span>systems <span class="_ _5"> </span>of <span class="_ _4"> </span>linear <span class="_ _4"> </span>equations. <span class="_ _4"> </span>Historically, <span class="_ _4"> </span>the <span class="_ _4"> </span>earl<span class="_ _6"></span>y <span class="_ _4"> </span>emphasis <span class="_ _4"> </span>was <span class="_ _4"> </span>on <span class="_ _4"> </span>the </div><div class="t m0 x1 h4 y8 ff1 fs1 fc0 sc0 ls0 ws0">determinant, not <span class="_ _6"></span>the matrix. In modern <span class="_ _6"></span>treatments <span class="_ _6"></span>of<span class="_"> </span>linear algebra, matrices <span class="_ _6"></span>are considered <span class="_ _6"></span>first. </div><div class="t m0 x1 h4 y9 ff1 fs1 fc0 sc0 ls0 ws0">We <span class="_ _6"></span>will <span class="_ _6"></span>not <span class="_ _6"></span>speculate <span class="_ _6"></span>much on <span class="_ _6"></span>this <span class="_ _6"></span>issue. <span class="_ _6"></span>Matrices <span class="_ _6"></span>provide <span class="_ _6"></span>a <span class="_ _6"></span>theoretically <span class="_ _6"></span>and <span class="_ _6"></span>practically <span class="_ _6"></span>useful </div><div class="t m0 x1 h4 ya ff1 fs1 fc0 sc0 ls0 ws0">way of approaching many<span class="_ _0"></span> t<span class="_ _6"></span>ype<span class="_ _0"></span>s of problems including:</div><div class="t m0 x2 h4 yb ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Solution of S<span class="_ _6"></span>y<span class="_ _8"></span>stems of Linear Equations,</span></div><div class="t m0 x2 h4 yc ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Equilibrium of Rigid Bodies (in physics),</span></div><div class="t m0 x2 h4 yd ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Graph Theory,</span></div><div class="t m0 x2 h4 ye ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Theory of Games,</span></div><div class="t m0 x2 h4 yf ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Leontief<span class="_"> </span>Economics<span class="_"> </span>Model,</span></div><div class="t m0 x2 h4 y10 ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Forest Management,</span></div><div class="t m0 x2 h4 y11 ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Computer Graphics, and Computed Tomography,</span></div><div class="t m0 x2 h4 y12 ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Genetics,</span></div><div class="t m0 x2 h4 y13 ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Cryptography,</span></div><div class="t m0 x2 h4 y14 ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Electrical Networks,</span></div><div class="t m0 x2 h4 y15 ff3 fs2 fc0 sc0 ls0 ws0">&#61623;<span class="_ _7"> </span><span class="ff1 fs1">Fractals.</span></div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>
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