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 = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 ${\mathcal {S}}$ .