53Linear-Algebra_opt.zip_linear algebra_matlab例程_数值算法/人工智能下载-pudn.com

<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 Mathematics</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:</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 and Determinants were discovered and developed in the eighteenth and </div><div class="t m0 x1 h4 y6 ff1 fs1 fc0 sc0 ls0 ws0">nineteenth centuries. Initially, their development dealt with transformation of geometric objects </div><div class="t m0 x1 h4 y7 ff1 fs1 fc0 sc0 ls0 ws0">and solution of systems of linear equations. Historically, the early emphasis was on the </div><div class="t m0 x1 h4 y8 ff1 fs1 fc0 sc0 ls0 ws0">determinant, not the matrix. In modern treatments of linear algebra, matrices are considered first. </div><div class="t m0 x1 h4 y9 ff1 fs1 fc0 sc0 ls0 ws0">We will not speculate much on this issue. Matrices provide a theoretically and practically useful </div><div class="t m0 x1 h4 ya ff1 fs1 fc0 sc0 ls0 ws0">way of approaching many types of problems including:</div><div class="t m0 x2 h4 yb ff3 fs2 fc0 sc0 ls0 ws0"> Solution of Systems of Linear Equations,</div><div class="t m0 x2 h4 yc ff3 fs2 fc0 sc0 ls0 ws0"> Equilibrium of Rigid Bodies (in physics),</div><div class="t m0 x2 h4 yd ff3 fs2 fc0 sc0 ls0 ws0"> Graph Theory,</div><div class="t m0 x2 h4 ye ff3 fs2 fc0 sc0 ls0 ws0"> Theory of Games,</div><div class="t m0 x2 h4 yf ff3 fs2 fc0 sc0 ls0 ws0"> Leontief Economics Model,</div><div class="t m0 x2 h4 y10 ff3 fs2 fc0 sc0 ls0 ws0"> Forest Management,</div><div class="t m0 x2 h4 y11 ff3 fs2 fc0 sc0 ls0 ws0"> Computer Graphics, and Computed Tomography,</div><div class="t m0 x2 h4 y12 ff3 fs2 fc0 sc0 ls0 ws0"> Genetics,</div><div class="t m0 x2 h4 y13 ff3 fs2 fc0 sc0 ls0 ws0"> Cryptography,</div><div class="t m0 x2 h4 y14 ff3 fs2 fc0 sc0 ls0 ws0"> Electrical Networks,</div><div class="t m0 x2 h4 y15 ff3 fs2 fc0 sc0 ls0 ws0"> Fractals.</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622b80033d2fbb00077d7016/bg2.jpg"><div class="t m0 x1 h3 y16 ff2 fs1 fc0 sc0 ls0 ws0">Introduction and Basic Operations:</div><div class="t m0 x1 h4 y17 ff1 fs1 fc0 sc0 ls0 ws0">Matrices, though they may appear weird objects at first, are a very important tool in expressing </div><div class="t m0 x1 h4 y18 ff1 fs1 fc0 sc0 ls0 ws0">and discussing problems which arise from real life cases.</div><div class="t m0 x1 h4 y19 ff1 fs1 fc0 sc0 ls0 ws0">Our first example deals with economics. Indeed, consider two families A and B (though we may </div><div class="t m0 x1 h4 y1a ff1 fs1 fc0 sc0 ls0 ws0">easily take more than two). Every month, the two families have expenses such as: utilities, </div><div class="t m0 x1 h4 y1b ff1 fs1 fc0 sc0 ls0 ws0">health, entertainment, food, etc... Let us restrict ourselves to: food, utilities, and health. How </div><div class="t m0 x1 h4 y1c ff1 fs1 fc0 sc0 ls0 ws0">would one represent the data collected? Many ways are available but one of them has an </div><div class="t m0 x1 h4 y1d ff1 fs1 fc0 sc0 ls0 ws0">advantage of combining the data so that it is easy to manipulate them. Indeed, we will write the </div><div class="t m0 x1 h4 y1e ff1 fs1 fc0 sc0 ls0 ws0">data as follows: </div><div class="t m0 x1 h4 y1f ff1 fs1 fc0 sc0 ls0 ws0">If we have no problem confusing the names and what the expenses are, then we may write </div><div class="t m0 x1 h4 y20 ff1 fs1 fc0 sc0 ls0 ws0">This is what we call a Matrix. The size of the matrix, as a block, is defined by the number of </div><div class="t m0 x1 h4 y21 ff1 fs1 fc0 sc0 ls0 ws0">Rows and the number of Columns. In this case, the above matrix has 2 rows and 3 columns. You </div><div class="t m0 x1 h4 y22 ff1 fs1 fc0 sc0 ls0 ws0">may easily come up with a matrix which has m rows and n columns. In this case, we say that the </div><div class="t m0 x1 h4 y23 ff1 fs1 fc0 sc0 ls0 ws0">matrix is a (mxn) matrix (pronounce m-by-n matrix). Keep in mind that the first entry (meaning </div><div class="t m0 x1 h4 y24 ff1 fs1 fc0 sc0 ls0 ws0">m) is the number of rows while the second entry (n) is the number of columns. Our above matrix </div><div class="t m0 x1 h4 y25 ff1 fs1 fc0 sc0 ls0 ws0">is a (2x3) matrix.</div><div class="t m0 x1 h4 y26 ff1 fs1 fc0 sc0 ls0 ws0">When the numbers of rows and columns are equal, we call the matrix a square matrix. A square </div><div class="t m0 x1 h4 y27 ff1 fs1 fc0 sc0 ls0 ws0">matrix of order n, is a (nxn) matrix.</div><div class="t m0 x1 h4 y28 ff1 fs1 fc0 sc0 ls0 ws0">Back to our example, let us assume, for example, that the matrices for the months of January, </div><div class="t m0 x1 h4 y29 ff1 fs1 fc0 sc0 ls0 ws0">February, and March are </div><div class="t m0 x1 h4 y2a ff1 fs1 fc0 sc0 ls0 ws0">To make sure that the reader knows what these numbers mean, you should be able to give the </div><div class="t m0 x1 h4 y2b ff1 fs1 fc0 sc0 ls0 ws0">Health-expenses for family A and Food-expenses for family B during the month of February. </div><div class="t m0 x1 h4 y2c ff1 fs1 fc0 sc0 ls0 ws0">The answers are 250 and 600. The next question may sound easy to answer, but requires a new </div><div class="t m0 x1 h4 y2d ff1 fs1 fc0 sc0 ls0 ws0">concept in the matrix context. Indeed, what is the matrix-expense for the two families for the first </div><div class="t m0 x1 h4 y2e ff1 fs1 fc0 sc0 ls0 ws0">quarter? The idea is to add the three matrices above. It is easy to determine the total expenses for </div><div class="t m0 x1 h4 y2f ff1 fs1 fc0 sc0 ls0 ws0">each family and each item, then the answer is </div><div class="t m0 x1 h4 y30 ff1 fs1 fc0 sc0 ls0 ws0">So how do we add matrices? An approach is given by the above example. The answer is to add </div><div class="t m0 x1 h4 y31 ff1 fs1 fc0 sc0 ls0 ws0">entries one by one. For example, we have </div><div class="t m0 x1 h4 y32 ff1 fs1 fc0 sc0 ls0 ws0">Clearly, if you want to double a matrix, it is enough to add the matrix to itself. So we have </div><div class="t m0 x1 h4 y33 ff1 fs1 fc0 sc0 ls0 ws0">we get</div><div class="t m0 x1 h4 y34 ff1 fs1 fc0 sc0 ls0 ws0">which implies </div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622b80033d2fbb00077d7016/bg3.jpg"><div class="t m0 x1 h4 y35 ff1 fs1 fc0 sc0 ls0 ws0">This suggests the following rule </div><div class="t m0 x1 h4 y17 ff1 fs1 fc0 sc0 ls0 ws0">and for any number , we will have </div><div class="t m0 x1 h4 y36 ff1 fs1 fc0 sc0 ls0 ws0">Let us summarize these two rules about matrices.</div><div class="t m0 x1 h4 y37 ff1 fs1 fc0 sc0 ls0 ws0">Addition of Matrices: In order to add two matrices, we add the entries one by one. </div><div class="t m0 x1 h4 y38 ff1 fs1 fc0 sc0 ls0 ws0">Note: Matrices involved in the addition operation must have the same size.</div><div class="t m0 x1 h4 y39 ff1 fs1 fc0 sc0 ls0 ws0">Multiplication of a Matrix by a Number: In order to multiply a matrix by a number, you multiply </div><div class="t m0 x1 h4 y3a ff1 fs1 fc0 sc0 ls0 ws0">every entry by the given number.</div><div class="t m0 x1 h4 y3b ff1 fs1 fc0 sc0 ls0 ws0">Keep in mind that we always write numbers to the left and matrices to the right (in the case of </div><div class="t m0 x1 h4 y3c ff1 fs1 fc0 sc0 ls0 ws0">multiplication).</div><div class="t m0 x1 h4 y3d ff1 fs1 fc0 sc0 ls0 ws0">What about subtracting two matrices? It is easy, since subtraction is a combination of the two </div><div class="t m0 x1 h4 y3e ff1 fs1 fc0 sc0 ls0 ws0">above rules. Indeed, if M and N are two matrices, then we will write </div><div class="t m0 x1 h4 y3f ff1 fs1 fc0 sc0 ls0 ws0">M-N = M + (-1)N</div><div class="t m0 x1 h4 y40 ff1 fs1 fc0 sc0 ls0 ws0">So first, you multiply the matrix N by -1, and then add the result to the matrix M.</div><div class="t m0 x1 h4 y41 ff1 fs1 fc0 sc0 ls0 ws0">Example. Consider the three matrices J, F, and M from above. Evaluate </div><div class="t m0 x1 h4 y42 ff1 fs1 fc0 sc0 ls0 ws0">Answer. We have </div><div class="t m0 x1 h4 y43 ff1 fs1 fc0 sc0 ls0 ws0">and since </div><div class="t m0 x1 h4 y44 ff1 fs1 fc0 sc0 ls0 ws0">we get </div><div class="t m0 x1 h4 y45 ff1 fs1 fc0 sc0 ls0 ws0">To compute J-M, we note first that </div><div class="t m0 x1 h4 y46 ff1 fs1 fc0 sc0 ls0 ws0">Since J-M = J + (-1)M, we get </div><div class="t m0 x1 h4 y47 ff1 fs1 fc0 sc0 ls0 ws0">And finally, for J-F+2M, we have a choice. Here we would like to emphasize the fact that </div><div class="t m0 x1 h4 y48 ff1 fs1 fc0 sc0 ls0 ws0">addition of matrices may involve more than one matrix. In this case, you may perform the </div><div class="t m0 x1 h4 y49 ff1 fs1 fc0 sc0 ls0 ws0">calculations in any order. This is called associativity of the operations. So first we will take care </div><div class="t m0 x1 h4 y4a ff1 fs1 fc0 sc0 ls0 ws0">of -F and 2M to get </div><div class="t m0 x1 h4 y4b ff1 fs1 fc0 sc0 ls0 ws0">Since J-F+2M = J + (-1)F + 2M, we get </div><div class="t m0 x1 h4 y4c ff1 fs1 fc0 sc0 ls0 ws0">So first we will evaluate J-F to get </div><div class="t m0 x1 h4 y4d ff1 fs1 fc0 sc0 ls0 ws0">to which we add 2M, to finally obtain </div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622b80033d2fbb00077d7016/bg4.jpg"><div class="t m0 x1 h4 y35 ff1 fs1 fc0 sc0 ls0 ws0">For the addition of matrices, one special matrix plays a role similar to the number zero. Indeed, if </div><div class="t m0 x1 h4 y4e ff1 fs1 fc0 sc0 ls0 ws0">we consider the matrix with all its entries equal to 0, then it is easy to check that this matrix has </div><div class="t m0 x1 h4 y4f ff1 fs1 fc0 sc0 ls0 ws0">behavior similar to the number zero. For example, we have </div><div class="t m0 x1 h4 y50 ff1 fs1 fc0 sc0 ls0 ws0">and </div><div class="t m0 x3 h5 y51 ff4 fs3 fc0 sc0 ls0 ws0">Matrices, though they may appear weird objects at first, are a very important tool in expressing and </div><div class="t m0 x1 h5 y52 ff4 fs3 fc0 sc0 ls0 ws0">discussing problems which arise from real life cases.</div><div class="t m0 x1 h4 y53 ff1 fs1 fc0 sc0 ls0 ws0">Our first example deals with economics. Indeed, consider twofamilies A and B (though we may </div><div class="t m0 x1 h4 y54 ff1 fs1 fc0 sc0 ls0 ws0">easily take more than two). Every month, the two families have expenses such as: utilities, </div><div class="t m0 x1 h4 y55 ff1 fs1 fc0 sc0 ls0 ws0">health, entertainment, food, etc... Let us restrict ourselves to: food, utilities, and health. How </div><div class="t m0 x1 h4 y56 ff1 fs1 fc0 sc0 ls0 ws0">would one represent the data collected? Many ways are available but one of them has </div><div class="t m0 x1 h4 y57 ff1 fs1 fc0 sc0 ls0 ws0">anadvantage of combining the data so that it is easy to manipulate them. Indeed, we will write </div><div class="t m0 x1 h4 y58 ff1 fs1 fc0 sc0 ls0 ws0">the data as follows:</div><div class="t m0 x1 h4 y59 ff1 fs1 fc0 sc0 ls0 ws0">If we have no problem confusing the names and what theexpenses are, then we may write</div><div class="t m0 x1 h4 y5a ff1 fs1 fc0 sc0 ls0 ws0">This is what we call a Matrix. The size of the matrix, as a block, is defined by </div><div class="t m0 x1 h4 y5b ff1 fs1 fc0 sc0 ls0 ws0">the number of Rows and the number ofColumns. In this case, the above matrix has 2 rows and 3 </div><div class="t m0 x1 h4 y5c ff1 fs1 fc0 sc0 ls0 ws0">columns. You may easily come up with a matrix which has m rows and n columns. In this case, </div><div class="t m0 x1 h4 y5d ff1 fs1 fc0 sc0 ls0 ws0">we say that the matrix is a(mxn) matrix (pronounce m-by-n matrix). Keep in mind that the first </div><div class="t m0 x1 h4 y5e ff1 fs1 fc0 sc0 ls0 ws0">entry (meaning m) is the number of rows while the second entry (n) is the number of columns. </div><div class="t m0 x1 h4 y5f ff1 fs1 fc0 sc0 ls0 ws0">Our above matrix is a (2x3) matrix.</div><div class="t m0 x1 h4 y60 ff1 fs1 fc0 sc0 ls0 ws0">When the numbers of rows and columns are equal, we call the matrix a square matrix. A square </div><div class="t m0 x1 h4 y61 ff1 fs1 fc0 sc0 ls0 ws0">matrix of order n, is a (nxn) matrix.</div><div class="t m0 x1 h4 y62 ff1 fs1 fc0 sc0 ls0 ws0">Back to our example, let us assume, for example, that the matrices for the months of January, </div><div class="t m0 x1 h4 y63 ff1 fs1 fc0 sc0 ls0 ws0">February, and March are</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622b80033d2fbb00077d7016/bg5.jpg"><div class="t m0 x1 h4 y64 ff1 fs1 fc0 sc0 ls0 ws0">To make sure that the reader knows what these numbersmean, you should be able to give the </div><div class="t m0 x1 h4 y65 ff1 fs1 fc0 sc0 ls0 ws0">Health-expenses for family A and Food-expenses for family B during the month of </div><div class="t m0 x1 h4 y66 ff1 fs1 fc0 sc0 ls0 ws0">February. The answers are 250 and 600. The next question may sound easy to answer, but </div><div class="t m0 x1 h4 y67 ff1 fs1 fc0 sc0 ls0 ws0">requires a new concept in the matrix context. Indeed, what is the matrix-expense for the </div><div class="t m0 x1 h4 y68 ff1 fs1 fc0 sc0 ls0 ws0">twofamilies for the first quarter? The idea is to add the three matrices above. It is easy to </div><div class="t m0 x1 h4 y69 ff1 fs1 fc0 sc0 ls0 ws0">determine the total expenses for each family and each item, then the answer is</div><div class="t m0 x1 h4 y6a ff1 fs1 fc0 sc0 ls0 ws0">So how do we add matrices? An approach is given by the above example. The answer is to add </div><div class="t m0 x1 h4 y6b ff1 fs1 fc0 sc0 ls0 ws0">entries one by one. For example, we have</div><div class="t m0 x1 h4 y6c ff1 fs1 fc0 sc0 ls0 ws0">Clearly, if you want to double a matrix, it is enough to add the matrix to itself. So we have</div><div class="t m0 x1 h4 y6d ff1 fs1 fc0 sc0 ls0 ws0">we get</div><div class="t m0 x1 h4 y6e ff1 fs1 fc0 sc0 ls0 ws0">which implies</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>