We define the span of a collection of vectors and explore the concept algebraically and geometrically.

### VEC-0090: Span

#### Linear Combinations Revisited

Recall that a vector is said to be a linear combination of vectors if for some scalars .

We write the system in augmented matrix form and apply elementary row operations to bring it to reduced row-echelon form.

This shows that and , and we can express as a linear combination of and as follows:

Observe that because vector is a linear combination of and , is the diagonal of a parallelogram whose sides are scalar multiples of and . As such, lies in the same plane as and , as illustrated below.

item:spanintro2 We need to solve the following vector equation: This equation corresponds to the system:

Writing the system in augmented matrix form and applying elementary row operations gives us the following reduced row-echelon form: We conclude that there are no solutions, and is not a linear combination of and .

Geometrically, this means that is not the diagonal of any parallelogram whose sides are scalar multiples of and . Thus, does not lie in the plane determined by and .

In part item:spanintro1 of Example ex:spanintro we expressed as a linear combination of and , and concluded
that lies in the plane determined by and . We will now introduce a new vocabulary
term and say that is *in the span* of and . In fact, every vector in the plane
determined by and is in the span of and . We say that and *span the
plane*.

In contrast, vector of part item:spanintro2 of Example ex:spanintro is not a linear combination of and . We say that is not in the span of and .

The following video takes another look at Example ex:spanintro using our new vocabulary.

#### Definition of Span

*span*of . We write and we say that vectors

*span*. Any vector in is said to be

*in the span*of . The set is called a

*spanning set*for .

Geometrically, we can use Procedure pro:lincombgeo of VEC-0040 to express any vector of as a linear combination of and . So, intuitively it makes sense that the two vectors span all of .

To verify this claim algebraically we will show that an arbitrary vector of can be written as a linear combination of and .

Consider the vector equation:

This corresponds to the system:

Writing the system in augmented matrix form and applying elementary row operations gives us the following reduced row-echelon form: This shows that every vector of can be written as a linear combination of and :

We conclude that

The span of and consists of elements of the form

Geometrically, we can interpret all such linear combinations as diagonals of parallelograms determined by scalar multiples of and . All such diagonals will lie in the plane determined by and . Let this plane be called . A portion of is shown below.

Because Procedure pro:lincombgeo of VEC-0040 can be applied to vectors that lie in just as easily as it can be applied to vectors of , we conclude that every vector in can be expressed as a linear combination of and . Thus,