Finite dimension vectore space and prove that exstence theoremfor bases
Answers
Answer:
Step-by-step explanation:
In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and defines the dimension of the vector space.
Formally, the dimension theorem for vector spaces states that
Given a vector space V, any two bases have the same cardinality.
As a basis is a generating set that is linearly independent, the theorem is a consequence of the following theorem, which is also useful:
In a vector space V, if G is a generating set, and I is a linearly independent set, then the cardinality of I is not larger than the cardinality of G.
In particular if V is finitely generated, then all its bases are finite and have the same number of elements.