Which statement defines a basis of a vector space?

A basis of a vector space is defined specifically by two key properties: linear independence and spanning. A set of vectors is considered linearly independent if none of the vectors in the set can be expressed as a linear combination of the others. This property ensures that the vectors provide unique directions in the vector space without redundancy.

Additionally, the set must span the vector space, meaning that any vector in the space can be expressed as a linear combination of the vectors in the basis. This ensures that the basis provides complete coverage of the space.

Therefore, the correct statement identifies these two crucial characteristics, making it clear that a basis is not just any set of vectors, but one that uniquely satisfies the conditions necessary to define the structure of the vector space fully. This understanding is fundamental in linear algebra, as it allows for a compact and efficient representation of the entire space.