06-240/Classnotes For Thursday, September 28

Linear Combination

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

a_i $\in$ F, $\sum _{i=1}^{n}$ a_iu_i = a₁u₁ + a₂u₂+ ... +a_nu_n

is called a "Linear Combination" of the u_i.

Span

span(u_i):= The set of all possible linear combinations of the u_i'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 = $\sum _{i=1}^{n}$ a_iu_i, u_i $\in {\mathcal {S}}$, y = $\sum _{i=1}^{m}$ b_iv_i, v_i $\in {\mathcal {S}}$. $\Rightarrow$ x+y = $\sum _{i=1}^{n}$ a_iu_i + $\sum _{i=1}^{m}$ b_iv_i = $\sum _{i=1}^{m+n}$ c_iw_i where c_i=(a₁, a₂,...,a_n, b₁, b₂,...,b_m) and w_i=c_i=(u₁, u₂,...,u_n, v₁, v₂,...,v_m).
3. cx= c$\sum _{i=1}^{n}$ a_iu_i=$\sum _{i=1}^{n}$ (ca_i)u_i$\in$ span ${\mathcal {S}}$.

Example 1. Let P₃($\Re$)={ax³+bx²+cx+d}$\subseteq$P($\Re$), 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)
$\Leftrightarrow$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, $\Rightarrow$ v $\in$ W.