11.Euclid空间,正交变换,对称变换.ppt

1 第十一讲第十一讲 Euclid Euclid 空间、正交变换和对称变换空间、正交变换和对称变换定义定义 1 1 设设 V V 是一个实线性空间是一个实线性空间, , 如果对如果对 V V 中任意的两个中任意的两个向量向量??????, , 都都有唯一的一个实数有唯一的一个实数 ( ( ??????) ) 与之对应与之对应, , 且满足且满足以下性质以下性质(1) (1) ????, , ???? V, ( V, ( ??, , ??) = ( ) = ( ??, , ??); ); (2) (2) ????, , ??, , ???? V, ( V, ( ??, , ??+ +??) = ( ) = ( ??, , ??) + ( ) + ( ??, , ??); ); (3) (3) ????, , ???? V, V, ??k k?? R, (k R, (k ??, , ??) = k( ) = k( ??, ,??); ); (4) (4) ?????? V, ( V, ( ??, , ??) ) ?? 0, 0, 且且( (??, , ??) = 0 ) = 0 ???? = 0, = 0, 则称则称??????????是向量是向量??与与??的内积的内积, , 定义了定义了内积内积的实线性的实线性空间空间 V V 称为欧几里得空间称为欧几里得空间, , 或简称欧氏空间或简称欧氏空间. . Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. 2 定义定义 2 2 设设?? 1 1, , ?? 2 2???????? n n 是是欧氏空间欧氏空间 V V 的一组基的一组基. . 对任意对任意??, , ???? V, V, 可设可设???????? 1 1, , ?? 2 2, ,??, , ?? n n??X X = = x x ??????+ +x x 2 2?? 2 2 ?????? x x n n?? n n, , ???????? 1 1, , ?? 2 2, ,??, , ?? n n??Y Y = = y y ??????+ +y y 2 2?? 2 2 ?????? y y n n?? n n, , 并计算并计算 1 1 1 1 ( , ) , ( , ) n n n n i i j j i j i j i j i j x y x y ?? ????? ? ??? ?? ?? ?? ?? ??? 1 1 1 2 1 1 2 1 2 2 2 2 1 2 1 2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , , , ) ( , ) ( , ) ( , ) nn n n n n n n yy x x x y ? ? ????? ?????? ?????? ???? ???? ?????