introduction.html (6431B)
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>introduction</title> 7 <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> 8 </head> 9 <body> 10 11 <div id="Introduction"><h2 id="Introduction">Introduction</h2></div> 12 <div id="Introduction-Linear Equations"><h3 id="Linear Equations">Linear Equations</h3></div> 13 <p> 14 "the study of linear equations" 15 </p> 16 17 <p> 18 a linear equation in the variables \(x_1, \dots, x_n\) has the form \(a_1 x_1+\dots+a_n x_n = b\), with \(a_1, \dots, a_n\) being the <em>coefficients</em> 19 </p> 20 21 <p> 22 geometric interpretation: 23 </p> 24 25 \[ 26 \begin{alignat*}{3} 27 &n=1\qquad &&a_1 x_1 = b \longrightarrow x_1 = \frac{b}{a_1}\qquad &&\text{(point on a line in $\Re$)}\\ 28 &n=2\qquad &&a_1 x_1 + a_2 x_2 = b \longrightarrow x_2 = \frac{b}{a_2} - \frac{a_1}{a_2}\qquad &&\text{(line in a plane in $\Re^2$)}\\ 29 &n=3\qquad &&a_1 x_1 + a_2 x_2 + a_3 x_3 = b\qquad &&\text{(planes in 3D space, in $\Re^3$)} 30 \end{alignat*} 31 \] 32 33 <p> 34 in general, \(n-1\)-dimensional planes in n-dimensional space 35 </p> 36 37 <p> 38 system of linear equations \(x_1, \dots, x_n\) is a collection of linear equations in these variables. 39 </p> 40 41 <p> 42 \(x_1 - 2x_2 = -1\) 43 </p> 44 45 <p> 46 \(-x_1 + 3x_2 = 3\) 47 </p> 48 49 <p> 50 If you graph them, you get this: 51 </p> 52 53 <p> 54 <img src="img/graph-example.png" alt="System of equations graph" /> 55 </p> 56 57 <p> 58 the solution is the intersection. 59 </p> 60 61 <p> 62 a system of linear equations has: 63 </p> 64 <ol> 65 <li> 66 no solutions (inconsistent) -- e.g. parallel lines 67 68 <li> 69 exactly 1 solution (consistent) 70 71 <li> 72 infinitely many solutions (consistent) - e.g. the same line twice 73 74 </ol> 75 76 <p> 77 two linear systems are "equivalent" if they have the same solutions. 78 </p> 79 80 <div id="Introduction-Matrix notation"><h3 id="Matrix notation">Matrix notation</h3></div> 81 <table> 82 <tr> 83 <th> 84 Equation 85 </th> 86 <th> 87 (Augmented) coefficient matrix notation 88 </th> 89 </tr> 90 <tr> 91 <td> 92 \(\begin{alignat*}{6} &x_1 &&-&&2x_2 &&+&&x_3 &&= 0\\ & && &&2x_2 &&-&&8x_3 &&= 8\\ &5x_1 && && &&-&&5x_3 &&= 10\end{alignat*}\) 93 </td> 94 <td> 95 \(\begin{bmatrix} 1 & -2 & 1 & 0\\ 0 & 2 & -8 & 8\\ 5 & 0 & -5 & 10 \end{bmatrix}\) 96 </td> 97 </tr> 98 </table> 99 100 <p> 101 the strategy to solve is to replace the system with an equivalent system that is easier to solve. 102 </p> 103 104 <p> 105 elementary row operations: 106 </p> 107 <ol> 108 <li> 109 replacement: add rows 110 111 <li> 112 scaling: multiply by constant (non-zero scalar) 113 114 <li> 115 interchange: swap two rows 116 117 </ol> 118 119 <p> 120 all of these are reversible & don't change the solution set. 121 </p> 122 123 <p> 124 Matrices A and B are equivalent (\(A \sim B\)) if there is a sequence of elementary operations to transform A to B. 125 </p> 126 127 <p> 128 If augmented matrices of two systems are row-equivalent, then the systems are equivalent. 129 </p> 130 131 <p> 132 Matrix A is in echelon form if: 133 </p> 134 <ol> 135 <li> 136 zero rows are below non-zero rows 137 138 <li> 139 the leading entry of a row is contained in a column that is to the left of the leading entry of the row below it. 140 141 </ol> 142 143 <p> 144 A is in reduced echelon form if: 145 </p> 146 <ol> 147 <li> 148 A is in echelon form 149 150 <li> 151 all leading entries are 1 152 153 <li> 154 the leading entry is the only non-zero entry in that column 155 156 </ol> 157 158 <div id="Introduction-Reducing a matrix"><h3 id="Reducing a matrix">Reducing a matrix</h3></div> 159 <p> 160 The reduced echelon form of a matrix is unique. 161 </p> 162 163 <p> 164 every matrix is row-equivalent to a unique reduced echelon matrix. 165 </p> 166 167 <p> 168 the positions of the leading entries in an echelon matrix are unique 169 </p> 170 171 <p> 172 \(\begin{bmatrix} \textbf{1} & * & * & *\\ 0 & 0 & \textbf{1} & *\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}\) 173 </p> 174 175 <p> 176 the values in bold are pivot positions. the columns containing those values are pivot columns. 177 </p> 178 179 <p> 180 Row reduction algorithm: 181 </p> 182 <ol> 183 <li> 184 Start with leftmost non-zero column (pivot column) 185 186 <li> 187 Select a non-zero entry as pivot and move it to the pivot position. 188 189 <li> 190 Create zeros below the pivot position. 191 192 <li> 193 Ignore the row containing the pivot position & repeat steps 1-3 until solved. The matrix will be in echelon form. 194 195 <li> 196 Make pivot positions equal to 1, create zeros in all pivot columns. Start with the rightmost column. The matrix will be in reduced echelon form. 197 198 </ol> 199 200 <p> 201 Side note: a computer chooses as pivot the entry that's smallest in absolute value to minimize the round-off error. 202 </p> 203 204 <p> 205 Basic variables correspond to pivot columns. Free variables correspond to non-pivot columns. You solve the equation by expressing basic variables in terms of free variables. 206 </p> 207 208 <p> 209 The matrix can be written in parametric form, example with \(x_3\) being a free variable: 210 </p> 211 212 <p> 213 \(\binom{x_1}{x_2} = \big\{ \binom{1}{4} + \binom{5}{-1} x_3 \;\rvert\; x_3 \in \Re \big\}\) 214 </p> 215 216 <p> 217 if there are any free variables, there are infinite solutions. 218 </p> 219 220 <div id="Introduction-Vectors"><h3 id="Vectors">Vectors</h3></div> 221 <p> 222 A vector is a line. If you have a vector in the form \(\begin{bmatrix} a\\b\end{bmatrix}\), you can draw it as an arrow from the origin ending at the point \((a,b)\). 223 </p> 224 225 <p> 226 To add vectors, add the individual cells together. 227 </p> 228 229 <p> 230 A vector equation \(a_1 x_1 + a_2 x_2 + \dots + a_n x_n = b\) has the same solution set as \(\begin{bmatrix} a_1 & a_2 & \dots & a_n & b \end{bmatrix}\). 231 </p> 232 233 <p> 234 When asked whether \(b\) is in \(\text{Span} \{v_1, \dots, v_p\}\), you have to check whether the augmented matrix \(\begin{bmatrix} v_1 & \dots & v_p & b \end{bmatrix}\) has a solution. 235 </p> 236 237 <p> 238 \(b\) is a linear combination of \(A\) if \(Ax = b\) has a solution. 239 </p> 240 241 <p> 242 The span is the set of all linear combinations of the vectors. 243 </p> 244 245 <p> 246 To calculate \(Ax\), if the number of columns in A is the same as the number of rows in x, you can follow the definition: 247 </p> 248 249 \[ 250 Ax = \begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix} \begin{bmatrix} x_1 \\ \dots \\ x_n \end{bmatrix} = x_1 a_1 + x_2 a_2 + \dots + x_n a_n 251 \] 252 253 <p> 254 You also have the rules (matrix A, vectors u and v, scalar c): 255 </p> 256 <ul> 257 <li> 258 \(A(u+v) = Au + Av\) 259 260 <li> 261 \(A(cu) = c(Au)\) 262 263 </ul> 264 265 </body> 266 </html>