线性映射确定的线性空间的直和分解及其应用

Direct sum decomposition of linear spaces determined by linear maps and applications

令R为实数域,A为一个m×n矩阵,φA,φAT分别为A确定的由R^(n)到R^(m)和R^(m)到R^(n)的线性映射,则φA,φAT分别确定Rn和Rm的一个直和分解,且这样的直和分解是具体可实现的.将该结果应用于任一n维线性空间V,则可实现任一线性变换σ确定的V的一个直和分解V=W⊕U,使得U为σ-不变子空间,且可同时实现以下目标:将V的一个子空间W的基扩充为V的一个基;当V为欧氏空间时,将一个子空间W的正交基扩充为V的一个正交基,并给出W的正交补子空间.

Letbe the field of real numbers,φA andφAT the linear maps from R^(n)to R^(m)and from R^(m) to R^(n) respectively,determined by an m×n matrix A.ThenφA andφAT give rise to a direct sum decomposition of Rn and Rm respectively,and such a decomposition can be practically realized.Applying this result to an arbitrary-dimensional linear space V,a direct sum decomposition V=W+U of V can be realized such that U is invariant whenever a linear transformationσof V is given.Moreover,let W be a subspace of V,then the following goals can be realized at the same time,namely extending a basis of W to a basis of V,extending an orthogonal basis of W to an orthogonal basis of V when V is Euclidian,and giving the orthogonal complement subspace of W.