数学のブログ

行列 1次変換 平面、直線の像

線形代数演習〈理工系の数学入門コース/演習 新装版〉 (浅野 功義(著)、大関 清太(著)、岩波書店)の第2章(行列)、2-2(1次変換)、問題4の解答を求めてみる。

1

x = 3 x 1 + 6 y 1 y = x 1 + 2 y 1
x - 3 y = 0 y = x 3

よって、 問題の1次変換は、平面上のすべての点をこの直線上に写す。

2

x = - 2 y + t
x = 3 ( - 2 y 1 + t ) + 6 y 1 = 3 t y = - 2 y 1 + t + 2 y 1 = t
y = x 3

よって問題の直線はこの直線に写される。

3

y = 1 - 2 x 3
x = 3 x 1 + 2 - 4 x 1 = - x 1 + 2 y = x 1 + 2 - 4 x 1 3 = 2 - x 1 3
x 1 = 2 - x
y = 2 - ( 2 - x ) 3 x - 3 y = 0

コード(Wolfram Language, Jupyter)

f[{x_, y_}] := {3x+6y, x+2y}

points = RandomReal[{-10, 10}, {50, 2}]

{{-9.62199, -8.09276}, {-9.04604, 5.06108}, {-8.90977, 6.44579}, {5.90997, -4.0726}, 
 
>   {0.575743, 6.27179}, {-8.29166, 1.7088}, {-1.44646, -4.91866}, {-3.44377, -4.33518}, 
 
>   {5.8507, -3.23918}, {-1.33965, -6.09769}, {-2.5755, 3.03251}, {0.213079, 7.48427}, 
 
>   {-2.31752, -3.86836}, {-4.35173, 1.54863}, {-8.76527, 7.71673}, {9.43672, 9.17134}, 
 
>   {-2.73881, 0.634215}, {-8.08724, -8.38681}, {-3.74907, -1.25128}, 
 
>   {-2.86821, -8.97813}, {-1.72033, 6.82746}, {-1.37109, 4.21313}, {5.03634, -3.11365}, 
 
>   {7.58242, 4.45733}, {1.5096, 3.41659}, {-7.42018, 6.64018}, {-7.08815, 7.60008}, 
 
>   {5.4159, -6.28246}, {1.48637, 4.39866}, {3.0266, -7.22956}, {-5.28119, -3.04786}, 
 
>   {0.973368, -0.371274}, {-0.0503731, 0.152904}, {-4.45989, -7.50631}, 
 
>   {-3.12997, 9.66232}, {1.24648, 4.04895}, {-2.72178, -9.02267}, {-8.93643, -7.18645}, 
 
>   {7.56869, 0.0500129}, {-8.93577, -1.23052}, {-0.62959, 8.88674}, 
 
>   {-6.18871, 0.159532}, {-2.98115, -1.27809}, {0.0654293, 0.832817}, 
 
>   {-0.731976, 7.7025}, {-3.8694, 5.8882}, {2.54529, -1.64238}, {5.75515, 6.94356}, 
 
>   {-3.78871, 1.77275}, {-3.72014, 9.02335}}
ListPlot[Table[f[point], {point, points}], PlotStyle -> PointSize[Medium]]
Output