Span
span(ui):= The set of all possible linear combinations of the ui's.
If
V is any subset,
span
|
:= The set of all linear combination of vectors in
|
|
=
|
even if
is empty.
Theorem: For any
V, span
is a subspace of V.
Proof:
1. 0
span
.
2. Let x
span
, Let x
span
,
x =
aiui, ui
, y =
bivi, vi
.
x+y =
aiui +
bivi =
ciwi where ci=(a1, a2,...,an, b1, b2,...,bm) and wi=ci=(u1, u2,...,un, v1, v2,...,vm).
3. cx= c
aiui=
(cai)ui
span
.