Suppose we have a set of vectors v1,v2,...,vNv1,v2,...,vN that lie in a
vector space VV. Given scalars α1,α2,...,αNα1,α2,...,αN,
observe that the linear combination

α
1
v
1
+
α
2
v
2
+
.
.
.
+
α
N
v
N
α
1
v
1
+
α
2
v
2
+
.
.
.
+
α
N
v
N

(1)
is also a vector in VV.

Let M⊂VM⊂V be a set of vectors in VV. The *span* of MM, written
span (M) span (M), is the set of all linear combinations of the vectors in MM.

v
1
=
1
1
0
,
v
2
=
0
1
0
.
v
1
=
1
1
0
,
v
2
=
0
1
0
.

(2)
span ({v1,v2})= span ({v1,v2})= the x1x2x1x2-plane, i.e., for any x1,x2x1,x2 we can
write x1=α1x1=α1 and x2=α1+α2x2=α1+α2 for some α1,α2∈Rα1,α2∈R.

V={f:f(t) is periodic with period 2π}V={f:f(t) is periodic with period 2π}, M={ejkt}k=-BBM={ejkt}k=-BB

span (M)= span (M)= periodic, bandlimited (to BB) functions, i.e., f(t)f(t) such that f(t)=∑Bk=-BcKejktf(t)=∑Bk=-BcKejkt for some c-B,c-B+1,...,c0,c1,...,cB∈Cc-B,c-B+1,...,c0,c1,...,cB∈C.