Span
Linear Combinations Revisited
Recall that a vector is said to be a linear combination of vectors if for some scalars .
If possible, express the given vector as a linear combination of
Interpret your results geometrically.
item:spanintro1 We need to find coefficients and such that . To do this we need to solve the vector
equation:
This equation translates into the following system:
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 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
Let be vectors in . The set of all linear combinations of is called the 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 .
Describe .
First, observe that and are not scalar multiples of each other.
Geometrically, we can use Procedure pro:lincombgeo to express any vector of as a linear combination of and , indicating 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
Describe .
First, observe that are not scalar multiples of each other.
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 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,