Difference between revisions of "06-240/Classnotes For Thursday, September 28"

From Drorbn
Jump to: navigation, search
(Span)
Line 33: Line 33:
  
 
''Example''
 
''Example''
1. Let P<sub>3</sub>(<math>\Re</math>)={ax<sup>3</sup>+bx<sup>2</sup>+cx+d}<math>\subseteq</math>P(<math>\Re</math>), ''a'', ''b'', ''c'', ''d'', <math>\in \Re</math>.
+
1. Let P<sub>3</sub>(<math>\Re</math>)={ax<sup>3</sup>+bx<sup>2</sup>+cx+d}<math>\subseteq</math>P(<math>\Re</math>), ''a'', ''b'', ''c'', ''d'', <math>\in \Re</math>.<BR>
''u''<sub>1</sub>=''x''<sup>3</sup>-2''x''<sup>2</sup>-5''x''-3
+
''u''<sub>1</sub>=''x''<sup>3</sup>-2''x''<sup>2</sup>-5''x''-3<BR>
''u''<sub>2</sub>=3''x''<sup>3</sup>-5''x''<sup>2</sup>-4''x''-9
+
''u''<sub>2</sub>=3''x''<sup>3</sup>-5''x''<sup>2</sup>-4''x''-9<BR>
''v''=2''x''<sup>3</sup>-2''x''<sup>2</sup>+12''x''-6
+
''v''=2''x''<sup>3</sup>-2''x''<sup>2</sup>+12''x''-6<BR>
 +
Let W=spab(''u''<sub>1</sub>, ''u''<sub>2</sub>), <BR>
 +
Does ''v'' <math> \in </math> W?<BR>
 +
''v'' is in W if ''v''=''a''<sub>1</sub>''u''<sub>1</sub>+''a''<sub>1</sub>''u''<sub>2</sub><br> for some ''a''<sub>1</sub>, ''a''<sub>2</sub> <math> \in \Re </math>.
 +
 
 +
If <math>\exists</math> ''a''<sub>1</sub>, ''a''<sub>2</sub> <math>\in \Re</math>, <br>
 +
{| border="0" cellpadding="0" cellspacing="0" align="center"
 +
|-
 +
|2''x''<sup>3</sup>-2''x''<sup>2</sup>+12''x''-6
 +
|= ''a''<sub>1</sub>(''x''-2''x''<sup>2</sup>-5''x''-3) + ''a''<sub>2</sub>(3''x''<sup>3</sup>-5''x''<sup>2</sup>-4''x''-9)
 +
|
 +
|-
 +
|
 +
|=(''a''<sub>1</sub>+3''a''<sub>2</sub>)''x''<sup>3</sup> + (-2''a''<sub>1</sub> -5''a''<sub>2</sub>)''x''<sup>2</sup> + (-5''a''<sub>1</sub>-4''a''<sub>2</sub>)''x'' + (-3''a''<sub>1</sub>-9''a''<sub>2</sub>)
 +
|
 +
|-
 +
|&nbsp;
 +
|
 +
|
 +
|-
 +
|<div align="right"><math>\Leftrightarrow</math>2</div>
 +
|=''a''<sub>1</sub>+3''a''<sub>2</sub>
 +
|
 +
|-
 +
|<div align="right">-2</div>
 +
|=-2''a''<sub>1</sub>-5''a''<sub>2</sub>
 +
|
 +
|-
 +
|<div align="right">12</div>
 +
|=-5''a''<sub>1</sub>-4''a''<sub>2</sub>
 +
|
 +
|-
 +
|<div align="right">-6</div>
 +
|=-3''a''<sub>1</sub>-9''a''<sub>2</sub>
 +
|
 +
|}
 +
Solve the four equations above and we will get ''a''<sub>1</sub>=-4 and ''a''<sub>2</sub>=2.<br>
 +
Check if ''a''<sub>1</sub>=-4 and ''a''<sub>2</sub>=2 hold for all the 4 equations.<br>
 +
Since it's hold, <math>\Rightarrow</math> ''v'' <math>\in</math> W.

Revision as of 20:59, 29 September 2006

Linear Combination

Definition: Let (ui) = (u1, u2, ..., un) be a sequence of vectors in V. A sum of the form

ai  \in F, \sum_{i=1}^n 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 \mathcal{S} \subseteq V is any subset,

span \mathcal{S} := The set of all linear combination of vectors in \mathcal{S}
=\left \{ \sum_{i=0}^n a_i u_i, a_i \in \mbox{F}, u_i \in \mathcal{S} \right \} \ni 0

even if \mathcal{S} is empty.

Theorem: For any \mathcal{S} \subseteq V, span \mathcal{S} is a subspace of V.

Proof:
1. 0  \in span \mathcal{S}.
2. Let x  \in span \mathcal{S}, Let x  \in span \mathcal{S}, \Rightarrow x = \sum_{i=1}^n aiui, ui  \in \mathcal{S}, y = \sum_{i=1}^m bivi, vi  \in \mathcal{S}. \Rightarrow x+y = \sum_{i=1}^n aiui + \sum_{i=1}^m bivi = \sum_{i=1}^{m+n} ciwi where ci=(a1, a2,...,an, b1, b2,...,bm) and wi=ci=(u1, u2,...,un, v1, v2,...,vm).
3. cx= c\sum_{i=1}^n aiui=\sum_{i=1}^n (cai)ui\in span \mathcal{S}.


Example 1. Let P3(\Re)={ax3+bx2+cx+d}\subseteqP(\Re), a, b, c, d, \in \Re.
u1=x3-2x2-5x-3
u2=3x3-5x2-4x-9
v=2x3-2x2+12x-6
Let W=spab(u1, u2),
Does v  \in W?
v is in W if v=a1u1+a1u2
for some a1, a2  \in \Re .

If \exists a1, a2 \in \Re,

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)
 
\Leftrightarrow2
=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, \Rightarrow v \in W.