Page 117 - A Course in Linear Algebra with Applications

4.2: Vector Spaces and Subspaces 101

We have to decide if there are real numbers c and d such
that cA + dB — C. To see what this entails, equate cor-
responding vector entries on both sides of the equation to
obtain
c - d = - 1
c + 2d = 5
4c + d = 6
Thus C belongs to < A, B > if and only if this linear system
is consistent. It is quickly seen that the linear system has the
(unique) solution c = 1, d = 2. Hence C = A + 2B, so that C
does belong to the subspace < A, B >.
What is the geometrical meaning of this conclusion? Re-
call that A, B and C can be represented by line segments in
3-dimensional space with a common initial point I, say IP,
IQ and IR. A typical vector in < A, B > can be expressed in
the form sA + tB with real numbers s and t . Now sA and
I
tB are representable by line segments parallel to P and IQ
respectively. We obtain a line segment that represents sA+tB
by applying the parallelogram law; clearly the resulting line
I
segment will lie in the plane determined by P and IQ. Con-
versely, it is not difficult to see that any line segment lying in
this plane represents a vector of the form sA + tB. Therefore
the vectors in the subspace < A, B > are those that can be
represented by line segments drawn from I lying in the plane
I
determined by P and IQ. What we have shown is that IR
lies in this plane.

Finitely generated vector spaces
A vector space V is said to be finitely generated if there
is a finite subset {vi, V2,..., v&} of V such that
V =< vi,v 2 ,...,v f c >,
that is to say, every vector in V is a linear combination of the
vectors i, V2,. • •, v^, and so has the form
v
CiVi + C 2 V 2 -\ h C kV k