What defines a basis in vector space?

A basis in a vector space is defined as a minimal spanning set, meaning it is a set of vectors that both spans the entire vector space and is linearly independent. This definition ensures that every vector in the space can be written uniquely as a linear combination of the basis vectors, providing a structured way to represent any vector in that space.

When we refer to a minimal spanning set, we emphasize that removing any vector from this set would result in a loss of the spanning property; in other words, the remaining vectors would no longer be able to represent the entire vector space. This highlights the importance of linear independence, as each vector contributes uniquely to the span of the space without being expressible as a combination of the others.

In other options, while a maximal spanning set might suggest a collection that spans the entire space, it could potentially include redundant vectors, which would violate the requirement of linear independence. A collection of vectors does not necessarily imply that they span the space or have any structure regarding overlap, and a set with no overlapping vectors alone does not guarantee that the vectors will span the vector space or remain independent. Thus, the essence of a basis is captured accurately by defining it as a minimal spanning set.