Decomposition Of Linear Spaces

This paper has four parts.we found that the finer levels of primary decomposi-tion and cyclic decomposition and Jordan canonical form and rational canonical form depend on different case in[6].If Combining primary decomposition and cyclic de-composition,you will get exact decomposition.Making primary decomposition on linear transformation A in linear space V like V =(?)Vi firstly,then make cyclic de-composition on Vi like Vi=(?)Vij,so V=(?)Vij.And in[6],to make Aij and A be the simplest form,author takes proper base in Vij.If liner transformation A have minimal polynomial m(?)=p1(?)r1p1(?)r2…p1(?)rs(s?2),and pi(?)is monic non-reducible and is different with each other,ri is positive integer(1 ? i ? s),and V = V1(?)V1(?)V2(?)…(?)Vs,and Vi=?er pi(?)ri = {? ? V|pi(A)ri? = 0},the minimal polynomial of Vi = ?er pi(?)ri = {? ? V| pi(A)ri? = 0}.Assume A be ma-trix with order n in field F,then we define A-base in vector space,and decompose Vi more subtly,so we can reinterpret Jordan canonical form and rational canonical form.The fourth part is the expectations of the results.