06-240/Classnotes For Thursday, September 28
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:
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 .
Example
1. Let P3()={ax3+bx2+cx+d}P(), a, b, c, d, .
u1=x3-2x2-5x-3
u2=3x3-5x2-4x-9
v=2x3-2x2+12x-6
Let W=spab(u1, u2),
Does v W?
v is in W if v=a1u1+a1u2
for some a1, a2 .
If a1, a2 ,
2x3-2x2+12x-6 | = a1(x-2x2-5x-3) + a2(3x3-5x2-4x-9) | |
=(a1+3a2)x3 + (-2a1 -5a2)x2 + (-5a1-4a2)x + (-3a1-9a2) | ||
2
|
=a1+3a2 | |
-2
|
=-2a1-5a2 | |
12
|
=-5a1-4a2 | |
-6
|
=-3a1-9a2 |
Solve the four equations above and we will get a1=-4 and a2=2.
Check if a1=-4 and a2=2 hold for all the 4 equations.
Since it's hold, v W.