有限个真子空间的并的进一步探讨

A Further Discussion on the Union of Finite Proper Subspaces

无限域上的有限维线性空间不能由有限个真子空间的并来覆盖,这是高等代数中一个重要的结论.本文在规定"真子空间的个数是极小的"前提下给出此结论的两种简洁的证明方法;并给出线性空间的不属于任意给定的有限个真子空间的一组基;最后利用此结论给出两道研究生入学考试试题的证明.

It is well known that the finite dimensional linear space over infinite field can not be covered by the union of finite proper subspace in advanced algebra.In this paper we firstly give two new concise proofs of this conclusion under the condition that the number of proper subspaces is minimal.Then we find a basis of the linear space whose elements do not belong to any given finite proper subspace.Finally,using this conclusion this paper gives the proofs of two problems in the entrance examination for graduate students.