欧式空间的一个推广

A Generalization of Euclid Space

欧式空间指出:若V是数域F上的一个n维线性空间,α1,α2,…,αn是V的一个基,那么对于V中的任意n个向量β1,β2,…,βn,恰有V的一个线性变换σ,使σ(αi)=βi(i=1,2,…,n);在欧式空间中把它给推广,即在一定的条件下,找到存在一个正交变换σ,使得σ(αi)=βi(i=1,2,…,m)成立的充分必要条件,并给出相关题目的证明。

Euclid Space says that if V is a n - liner space in number field F and a1, a2,……, an is one of ba- sis of V, there will exactly exist a liner σ transformation such that a（ ai ） =βi for all i from to n to arbitrary vector β1 ,β2,…… ,βn. To generalize it, this paper not only tries to find the sufficient and necessary condition of an orthogo- nal transformation σ such that σ （ai） = βi for all i from to n under some certain circumstances but also provides proofs to a few relevant problems.