Which of these pairs is a requirement for a set to be a basis for a vector space?

For a set to be considered a basis for a vector space, it must meet two crucial conditions: it must be linearly independent and it must span the vector space.

Linear independence means that no vector in the set can be expressed as a linear combination of the others. This ensures that every vector in the basis contributes uniquely to the vector space, providing the necessary "building blocks" for the space.

Spanning the vector space means that every vector in the space can be formed as a linear combination of the basis vectors. In other words, the basis must be able to represent every vector in the vector space.

Together, these conditions guarantee that the basis is a minimal set of vectors that can describe the entire vector space without redundancy.

Other options do not fulfill the requirements for a basis. For instance, simply having three vectors, having a maximum number, or including zero vectors does not ensure that a set forms a basis since those characteristics do not inherently relate to independence or spanning properties.