关于欧几里得子空间正交补的正交基

关于欧式空间子空间正交补的正交基的计算,本文提出比schmidt process简单的一种算法。在n维欧式空间中,我们取一线性无关组M={α1,…,αm},m

On the Orthogonal Basis of the Orthogonal Complement of a Subspace in Euclidean Space

In Euclidean space En let M={α1, …, αm}, m〈n, be linearly independent, by elimination we obtain β1⊥M and construct M1={α1, …, αm, β1}. Using the above method successively we obtain： Mn-m={α1, …, αm, β1, …, βn-m}. Clearly the subspace T=〈β1, …, βN-M〉 is the orthogonal complement of the subspace S=〈α1, …, αm〉 , and {β1, …, βn-m} is the orthogonal basis of T. In this way we give a method called the process of iteration elimination which is simpler than Schmidt process on this problem. In particular when m=1, we give a simple formula, writing the orthogonal basis directly.