In terms of vector spaces, what does it mean to span?

The concept of spanning in the context of vector spaces refers to the ability of a set of vectors to generate every vector in the space through linear combinations. When a set of vectors spans a vector space, it means that any vector within that space can be expressed as a combination of those spanning vectors, using scalar multiplication and vector addition.

Choosing a set of vectors that spans a space is crucial because it enables us to understand the whole space in terms of these vectors, providing a foundation for the structure of the space. For instance, in a three-dimensional space, three non-coplanar vectors can serve as a spanning set, meaning any vector within that three-dimensional space can be represented as a combination of these three vectors.

This understanding reflects why the correct answer asserts that spanning refers to the creation of vectors capable of generating all others within the space, identifying the essence of spanning sets in linear algebra.