1. The problem is to define the span of a vector space. 2. The span of a set of vectors is the collection of all possible linear combinations of those vectors. 3. If you have vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$ in a vector space, the span is all vectors of the form: $$a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n \mathbf{v}_n$$ where $a_1, a_2, \ldots, a_n$ are scalars. 4. This means the span is the smallest subspace containing all the vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$. 5. In simple terms, the span is all the vectors you can reach by stretching and adding the given vectors together. 6. The span is important because it tells us how vectors can combine to fill a space or subspace. Final answer: The span of a set of vectors is the set of all linear combinations of those vectors, forming the smallest subspace containing them.