What does a spanning set of linearly independent vectors form?

A spanning set of linearly independent vectors forms a basis for a vector space. In linear algebra, a basis is defined as a set of vectors that is both linearly independent and spans the entire vector space.

When vectors are linearly independent, no vector in the set can be expressed as a linear combination of the others, which means they contribute uniquely to the span. Since they also span the space, this means any vector in that vector space can be expressed as a linear combination of the vectors in the basis. Therefore, a set of linearly independent vectors that spans a space satisfies both criteria for being a basis.

This concept is fundamental in linear algebra as it allows us to describe a vector space in terms of a minimal set of vectors, facilitating understanding and computation within that space. The other options do not fulfill the criteria required to define a basis: subsets of vectors do not necessarily span or maintain independence, null spaces refer to the set of vectors that map to zero under a linear transformation, and linear transformations describe functions between vector spaces rather than sets of vectors.