Span

Linear Combinations Revisited

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

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 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.

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Definition of Span

Practice Problems

Choose the best description for each set below.
Plane in Line in Line in
Plane in Line in Line in
Plane in Line in Line in
Which of the following pairs of sets are equal?
Let . Give an example of at least one vector such that , do NOT span a plane in . Describe .
Prove or disprove. The zero vector of is contained in the span of any collection of vectors of .