What does the span of a set of vectors represent?

The span of a set of vectors represents the set of all possible linear combinations of those vectors. This includes any vector that can be formed by multiplying each vector in the set by a scalar (which can be any real number) and then adding the results together.

Understanding this concept is crucial because it highlights how a small number of vectors can generate a larger vector space through combinations. For instance, if you have two vectors in a two-dimensional space, their span will cover the entire plane if they are not collinear. This relationship illustrates the concept of vector spaces and their dimensions effectively.

The other options do not accurately reflect the definition of span. While the individual vectors are part of the span, the span encompasses much more than just those original vectors. Similarly, while the maximum dimension of the vector space is related to linear independence and spans, it does not define what a span is. Finally, the orthogonal complement of a set of vectors refers to a different concept that deals with vectors being perpendicular to the original vectors in the set, which is not related to the span itself.