PC用は別頁
※旧教育課程の高校数学Cに含まれていた「1次変換」について,このサイトには次の教材があります.
この頁へGoogleやYAHOO ! などの検索から直接来てしまったので「前提となっている内容が分からない」という場合や「この頁は分かったがもっと応用問題を見たい」という場合は,他の頁を見てください.  が現在地です.
1次変換とは-現在地
点の像と原像
2点とその像→変換式
回転移動の1次変換
合成変換と逆変換
直線の像,領域の像

== 1次変換 ==

【1次変換とは】
■集合から集合への対応 f:A→B のうち,f:A→A のように,同一集合への対応を変換といいます。

■x,y平面上の点からx,y平面上の点への対応

 f:(x,y)(x’,y’) 
のうち,
x’=ax+by
y’=cx+dy
のように,x’,y’が定数項のない1次式で表されるものを1次変換と呼ばれます。

■1次変換は,行列を用いて

のように表すことができます。
※ 細かな区別ですが,→ と  は,使う場面によって,集合→集合,要素要素 のように使い分けていますが,この区別は高校ではあまり重視しません。

【1次変換の例】
■1 
x軸に関する対称移動
(x,y)(x,-y) 
すなわち
x’=x  
y’=-y 
と対応するので
x’=1・x+ 0・y
y’=0・x+(-1)・y
x軸に関する対称移動を表す1次変換の行列は
■2
y軸に関する対称移動
(x,y)(-x,y)
すなわち
x’=-x  
y’=y 
と対応するので
x’=(-1)・x+ 0・y
y’=0・x+1・y
y軸に関する対称移動を表す1次変換の行列は
■3
原点に関する対称移動
(x,y)(-x,-y)
すなわち
x’=-x  
y’=-y 
と対応するので
x’=(-1)・x+ 0・y
y’=0・x+(-1)・y
原点に関する対称移動を表す1次変換の行列は
■4
y=xの直線に関する対称移動
(授業では,この部分の誤解が特に多いので注意 「y=x」という言葉に惑わされることなく,図で考えることが大切)
(x,y)(y,x)
すなわち
x’=y
y’=x
と対応するので
x’=0・x+1・y
y’=1・x+0・y
y=xの直線に関する対称移動を表す1次変換の行列は
■5
y=-xの直線に関する対称移動
(x,y)(-y,-x)
すなわち
x’=-y  
y’=-x 
と対応するので
x’=0・x+(-1)・y
y’=(-1)・x+0・y
y=-xの直線に関する対称移動を表す1次変換の行列は
■6
恒等変換すなわち各点を自分自身に対応させる変換は,
(x,y)(x,y)
すなわち
x’=x
y’=y
と対応するので
x’=1・x+0・y
y’=0・x+1・y
恒等変換を表す1次変換の行列は
E=

(簡単)確認テスト
次の行列が表す1次変換を右から選びなさ
い。(初めに問題を選び,次に選択肢を選びなさい。)
問題


選択肢



[注]直前にPC版から入られた場合は,自動転送でスマホ版に来ていますので,ブラウザの[戻るキー]では戻れません(堂々巡りになる).下記のリンクを使ってメニューに戻ってください.
...(携帯版)メニューに戻る

...(PC版)メニューに戻る

■[個別の頁からの質問に対する回答][1次変換について/17.4.9]
とても良い!
=>[作者]:連絡ありがとう.この教材は,実際には体験・入門レベルの内容ですが,この教材でよい評価が得られる理由として,教育課程の改訂とか科目選択の都合で,そもそも習っていない方が読んでいる場合があるのかなと真剣に考えています.
■[個別の頁からの質問に対する回答][1次変換について/17.2.22]
東工大受験3日前に良い勉強できました。回転座標の問題や明らかに行列式をモチーフにした問題が出たら是非複素平面といった回りくどい方法を取らずにバッチリ一次変換を使って行きたいと思います
=>[作者]:連絡ありがとう.
■[個別の頁からの質問に対する回答][1次変換について/17.2.21]
とてもわかりやすいです。参考になりました。
=>[作者]:連絡ありがとう.

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