The span of a bunch of vectors is the set of all linear combinations of that bunch of vectors. We denote it as span(v_1, \dots v_{m)}. constituents for constructing a linear combination a list of vectors v_1,\dots,v_{m} and scalars a_1, a_2, \dots, a_{m} \in \mathbb{F} requirements \begin{equation} span(v_{1}..v_{m}) = {a_1v_1+\dots +a_{m}v_{m}:a_1\dots a_{m} \in \mathbb{F}} \end{equation} additional information span is the smallest subspace containing all vectors in the list Part 1: that a span of a list of vectors is a subspace containing those vectors By taking all a_{n} as 0, we show that the additive identity exists. Taking two linear combinations and adding them (i.e. adding two members of the span) is still in the span by commutativity and distributivity (reorganize each constant a_{1} together)—creating another linear combination and therefore a member of the span. Scaling a linear combination, by distributivity, just scales the scalars and create yet another linear combination. Part 2: a subspace containing the list of vectors contain the span subspaces are closed under scalar multiplication and addition. Therefore, we can just construct every linear combination. By double-containment, a subspace is the smallest subspace containing all vectors. \blacksquare spans If span(v_1, \dots v_{m}) equals V, we say that v_1, \dots, v_{m} spans V. NOTE! the two things have to be equal—if the span of a set of vectors is larger than V, they do not span V. length of linearly-independent list \leq length of spanning list see here.