欧氏空间的子空间的正交系与正交补

The Orthocomplement and the Orthosystem of a Subspace in the Euclidean Space

提出了欧氏空间的子空间的正交系概念,指出欧氏空间的子空间的正交系是唯一存在的,给出了利用欧氏空间的基的度量矩阵和齐次线性方程组的基础解系求欧氏空间的子空间的正交系的方法.明确了欧氏空间的子空间的正交系与正交补的区别和联系.指出欧氏空间的子空间如果有正交补,那么正交补和正交系是相同的;欧氏空间的有限维子空间都有正交补,无限维子空间不一定有正交补.

In this paper,the concept of orthosystem of a subspace in Euclidean space is presented.The unique existence of orthosystem of a subspace in the Euclidean space is pointed out.A method to obtain the orthosystem of a subspace in the Euclidean space is provided,using the metric matrix of a base in Euclidean space and a basic system of solutions in a system of homogeneous linear equations.The differences and the relation between the orthocomplement and the orthosystem of a subspace in the Euclidean space are stated clearly.It shows that:the orthocomplement and the orthosystem are the same collection of vectors if there is a orthocomplement of the subspace in the Euclidean space;there is a orthocomplement in every finite-dimensional subspace in the Euclidean space and may not necessarily be one in an infinite-dimensional subspace in the Euclidean space.