Linear Combination
Definition: Let (ui) = (u1, u2, ..., un) be a sequence of vectors in V. A sum of the form
- ai
F,
aiui = a1u1 + a2u2+ ... +anun
is called a "Linear Combination" of the ui.
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:
- 0
span
,
- 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).# cx= c
aiui=
(cai)ui
span
.