linear-transformations.html (2709B)
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>linear-transformations</title> 7 <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> 8 </head> 9 <body> 10 11 <div id="Linear transformations"><h2 id="Linear transformations">Linear transformations</h2></div> 12 <p> 13 definitions: 14 </p> 15 <ul> 16 <li> 17 transformation, function, mapping: rule assigning to each vector in \(\Re^n\) a vector \(T(x)\) in \(\Re^m\) 18 19 <li> 20 domain: set \(\Re^n\) 21 22 <li> 23 codomain: set \(\Re^m\) 24 25 <li> 26 image: vector T(x) 27 28 <li> 29 range: set of all images T(x) 30 31 </ul> 32 33 <p> 34 a projection transformation happens if you go to a lower dimension (e.g. \(x_3\) becomes 0). a shear transformation happens if a 2D square is tilted sideways into a parallelogram. 35 </p> 36 37 <p> 38 a transformation T is linear if: 39 </p> 40 <ol> 41 <li> 42 \(T(u + v) = T(u) + T(v)\) for all \(u,v \in \text{Domain}(T)\) 43 44 <li> 45 \(T(cu) = cT(u)\) for all scalars c and all \(u \in \text{Domain}(T)\) 46 47 </ol> 48 49 <p> 50 linear transformations preserve operations of vector addition and scalar multiplication. 51 </p> 52 53 <p> 54 if T is a linear transformation, then: 55 </p> 56 <ul> 57 <li> 58 \(T(0) = 0)\) 59 60 <li> 61 \(T(cu + dv) = cT(u) + dT(v)\) 62 63 <li> 64 \(T(c_1 v_2 + \dots + c_p v_p) = c_1 T(v_1) + \dots + c_p T(v_p)\) (superposition principle) 65 66 </ul> 67 68 <p> 69 given scalar r, and \(T: \Re^2 \rightarrow \Re^2\) by \(T(x) = rx\) 70 </p> 71 <ul> 72 <li> 73 contraction: when \(0 \leq r \leq 1\) 74 75 <li> 76 dilation: when \(r > 1\) 77 78 </ul> 79 80 <p> 81 every linear transformation \(\Re^n \rightarrow \Re^m\) is a matrix transformation \(x \mapsto Ax\). 82 </p> 83 84 <p> 85 \(A = [[T(e_1) \dots T(e_n)]\), where \(e_j\) is the jth column of the identity matrix in \(\Re^n\) 86 </p> 87 88 <p> 89 geometric linear transformations of \(\Re^2\): 90 </p> 91 92 <p> 93 <img src="img/geo-reflections.png" alt="Reflections" /> <img src="img/geo-contract-shears.png" alt="Contractions/expansions and shears" /> <img src="img/geo-projections.png" alt="Projections" /> 94 </p> 95 96 <p> 97 types of mappings: 98 </p> 99 <ul> 100 <li> 101 \(T: \Re^n \rightarrow \Re^m\) is 'onto' \(\Re^m\) if <em>each</em> b in \(\Re^m\) is the image of <em>at least one</em> x in \(\Re^n\). 102 103 <li> 104 \(T: \Re^n \rightarrow \Re^m\) is one-to-one if <em>each</em> b in \(\Re^m\) is the image of <em>max one</em> x in \(\Re^n\). 105 106 <ul> 107 <li> 108 so if \(T(x) = 0\) only has the trivial solution 109 110 </ul> 111 </ul> 112 113 <p> 114 for mapping \(T: \Re^n \rightarrow \Re^m\) and standard matrix \(A\): 115 </p> 116 <ul> 117 <li> 118 T maps \(\Re^n\) onto \(\Re^m\) iff columns of matrix span \(\Re^m\) 119 120 <li> 121 T is one-to-one iff columns of matrix are linearly independent. 122 123 </ul> 124 125 </body> 126 </html>