Page 116 - A Course in Linear Algebra with Applications

100 Chapter Four: Introduction to Vector Spaces

In general let X be any non-empty subset of a vector
space V and denote by

<X >
the set of all linear combinations of vectors in X. Thus a typical
element of < X > is a vector of the form

cixi + c 2 x 2 H h CfcXfc

where i, x 2 ,..., x& are vectors belonging to X and c\, c 2 ,...,
x
Ck are scalars. From this formula it is clear that the sum of
any two elements of < X > is still in < X > and that a scalar
multiple of an element of < X > is in < X >. Thus we have
the following important result.

Theorem 4.2.2
If X is a non-empty subset of a vector space V, then < X >,
the set of all linear combinations of elements of X, is a sub-
space of V.

We refer to < X > as the subspace of V generated (or
spanned) by X. A good way to think of < X > is as the small- -
est subspace of V that contains X. For any subspace of V that
contains X will necessarily contain all linear combinations of
vectors in X and so must contain < X > as a subset. In par-
ticular, a subset X is a subspace if and only if X — < X >.
x
In the case of a finite set X = {x!, x 2 ,..., /J, we shall write
< X i , X 2 , . . . , Xfc >
for < X >.

Example 4.2.4
For the three vectors of R 3 given below, determine whether
C belongs to the subspace generated by A and B:
x ,j, ,c
-(i) -(1) -(~i)-