solution-sets-of-linear-systems.html (3023B)
1 <!DOCTYPE html> 2 <html> 3 <head> 4 <script type="text/javascript" async src="https://cdn.jsdelivr.net/gh/mathjax/MathJax@2.7.5/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> 5 <link rel="Stylesheet" type="text/css" href="style.css"> 6 <title>solution-sets-of-linear-systems</title> 7 <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> 8 </head> 9 <body> 10 11 <div id="Solution sets of linear systems"><h2 id="Solution sets of linear systems">Solution sets of linear systems</h2></div> 12 <div id="Solution sets of linear systems-Homogeneous linear systems"><h3 id="Homogeneous linear systems">Homogeneous linear systems</h3></div> 13 <p> 14 homogeneous: if you can write it in \(Ax = 0\) where A is an \(m \times n\) matrix and 0 is the zero vector in \(\Re^m\) 15 </p> 16 <ul> 17 <li> 18 always has at least one solution (the trivial solution, \(x = 0\)). 19 20 <li> 21 has a nontrivial solution iff there is a free variable 22 23 <ul> 24 <li> 25 if the equation has only one free variable, the solution is a line through the origin 26 27 <li> 28 when there are two or more free variables, it's a line through the origin 29 30 </ul> 31 <li> 32 solution set is \(\text{Span} \{v_1, \ldots, v_p\}\) for suitable vectors 33 34 </ul> 35 36 <div id="Solution sets of linear systems-Parametric vector form"><h3 id="Parametric vector form">Parametric vector form</h3></div> 37 <p> 38 implicit description: 39 </p> 40 <ul> 41 <li> 42 a simple equation 43 44 <li> 45 e.g. \(10x_1 - 3x_2 - 2x_3 = 0\) 46 47 </ul> 48 49 <p> 50 explicit description (parametric vector form): 51 </p> 52 <ul> 53 <li> 54 the solution to the equation as a set spanned by u and v 55 56 <li> 57 of the form \(x = su + tv\), with \(s,t \in \Re\) 58 59 </ul> 60 61 <p> 62 the solution set of \(Ax = 0\) is \(x = tv\) with \(t \in \Re\). 63 </p> 64 65 <p> 66 if \(Ax = b\) has a solution, then you get the solution set by translating the solution set of \(Ax = 0\) using any particular solution p of \(Ax = b\). The set is then \(x = p + tv\) 67 </p> 68 69 <p> 70 Writing a solution set in parametric vector form: 71 </p> 72 <ol> 73 <li> 74 Row reduce augmented matrix to echelon form 75 76 <li> 77 Express each basic variable in terms of any free variables. 78 79 <li> 80 Write a typical solution x as a vector, with entries depending on the (potential) free variables. 81 82 <li> 83 Decompose x into a linear combination of vectors using free vars as parameters. 84 85 </ol> 86 87 <div id="Solution sets of linear systems-Linear independence"><h3 id="Linear independence">Linear independence</h3></div> 88 89 <p> 90 linearly independent: 91 </p> 92 <ul> 93 <li> 94 set of vector equations: iff the vector equation has only the trivial solution (\(x_1 = x_2 = x_3 = 0\)) 95 96 <li> 97 columns of matrix: iff \(Ax = 0\) has <em>only</em> the trivial solution 98 99 <li> 100 one vector: iff v is not the zero vector 101 102 <li> 103 two vectors: if neither of the vectors is a multiple of the other 104 105 </ul> 106 107 <p> 108 linearly dependent: 109 </p> 110 <ul> 111 <li> 112 iff at least one of the vectors is a linear combination of the others 113 114 <li> 115 if there are more vectors than entries in each vector 116 117 <li> 118 if the set contains the zero vector 119 120 </ul> 121 122 <p> 123 a set is linearly dependent iff it's not linearly independent. 124 </p> 125 126 </body> 127 </html>