span (linear algebra)

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 \(a_1, a_2, \dots, a_{m} \in \mathbb{F}\)

\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}

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. \(\blacksquare\)

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 \(V\).