Spanning Sets

Let be a set of vectors in a vector space .

The span of is

(see set builder notation)

That is, the span is the set of all linear combinations of the vectors, so the span consists of every possible vector that can be created by the vectors in the span.

We state that is spanned by and that is a spanning set for

Recall that the span is a subspace. Using the basis vectors in , , span , and we can create any vector in (with its tail on the origin) with these two vectors. Similarly, in , the basis vectors span .

Recall that we can chose any vectors to be our basis vectors, so long as each of these vectors is "unique" (cannot be expressed as a linear combination of the others).

These vectors span

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These vectors span

A spanning set is infinite, since it's for any in real

Let , , and .
if and only if the system is consistent

To see if , we only need to verify that the system is consistent, which means we need to carry the system to REF and apply the [Math/Linear Algebra/Matrices/Rank#System-Rank Theorem|system-rank theorem].

If we wish to write as a linear combination of , then we must solve the system for

Since and are not scalar multiples, they this represents the vector equation of a plane through the origin.

Let . One of these vectors, say , can be expressed as a linear combination of if and only if:

That is, if we can express any element of a spanning set as a linear combination of the other vectors, then that vector can be removed from the spanning set, and vice versa.

The following vectors span only

since the green vector can be expressed as a linear combination of the blue and red vectors, and is hence redundant. We can create it with the other two anyways.

Given

Since , we can remove from
Since , we can remove from

Since neither of the vectors are scalar multiples of each other, we cannot remove either of them without changing the span.