Is the span of a set of vectors considered closed?

The span of a set of vectors is defined as the set of all possible linear combinations of those vectors. This includes all combinations formed by taking any vector in the set and multiplying it by a scalar, followed by the addition of any other vectors (also scaled by a scalar).

The key aspect of the span is that it satisfies the closure properties of vector spaces. It is closed under both vector addition and scalar multiplication. This means that if you take any two vectors from the span and add them together, the resulting vector will also be in the span. Similarly, if you multiply any vector from the span by a scalar, the resulting vector will also be in the span.

Thus, the span of a set of vectors is indeed closed under both addition and scalar multiplication, making the correct identification of this property vital for a deep understanding of linear algebra concepts and their applications.