A finite vector space is a mathematical structure that consists of a set of objects, known as vectors, and a set of operations that can be performed on those vectors. The operations include addition and multiplication by scalars, and the resulting vectors must still be within the original set.
A set of vectors is said to be linearly independent if none of the vectors can be expressed as a linear combination of the others. In other words, no vector in the set is redundant and they all contribute unique information to the space.
The existence of a basis for a finite vector space with a linearly independent set can be proven using a theorem known as the Steinitz Exchange Lemma. This theorem states that any linearly independent set in a vector space can be expanded to form a basis by adding linearly independent vectors from the space.
A basis is a fundamental concept in linear algebra and serves as a building block for many other mathematical concepts and applications. Proving the existence of a basis ensures that the vector space is well-defined and has a solid foundation for further analysis and manipulation.
Yes, a finite vector space can have more than one basis. This is because there can be multiple sets of linearly independent vectors that span the same space. However, all bases for a given vector space will have the same number of elements, known as the dimension of the space.