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The polynomials 1, span the vector space defined in Example 5 since each polynomial p in can be written
as
which is a linear combination of 1, . We can denote this by writing
EXAMPLE 12 Three Vectors That Do Not Span span the vector space .
Determine whether , , and
Solution in can be expressed as a linear combination
We must determine whether an arbitrary vector
of the vectors , , and . Expressing this equation in terms of components gives
or
or
The problem thus reduces to determining whether this system is consistent for all values of , , and . By parts (e) and
(g) of Theorem 4.3.4, this system is consistent for all , , and if and only if the coefficient matrix
has a nonzero determinant. However, (verify), so , , and do not span .
Spanning sets are not unique. For example, any two noncollinear vectors that lie in the plane shown in Figure 5.2.5 will span
that same plane, and any nonzero vector on the line in that figure will span the same line. We leave the proof of the
following useful theorem as an exercise.
THEOREM 5.2.4
If and are two sets of vectors in a vector space V, then
if and only if each vector in S is a linear combination of those in and each vector in is a
