Weak Base and independent weak base of vector space

Abstract

The problem of generation and oneness considered for expressing about an element is very important and has a big effect in mathematics in general and in algebra in special for example in vector spaces which has a finite dimension,  where every element from this space is written in a lonely form in terms of elements of subset in this space and in this case we called this subset from the space (generated and linearly independent) base of space.

In second section of this paper we study the weak base for vector space, we obtained a full description weak dimension vector space and it is proved that finite weak bases for vector space have the same cardinality. In addition to that, it is proved that a finite subset  of vector space is weak base if and only if  is minimal weak generated set.

Also, we proved that every generated set of vector space contains weak base of this space. It is proved that for weak dimension spaces every weak independent subset can be extended to weak base for this space.

In addition to that, we obtain many of important and interesting properties for weak base. In third section we study the independent weak bases for vector space and we proved that every subspace of some space is weak generated by subset not contained in it, contains weak independent base.

Finally, we proved the sufficient and necessary condition to be subspace maximal in vector space , that every weak independent base of subspace is base of . Also, we proved that many important and interesting properties for weak independent bases of vector space.