06-240/Classnotes For Thursday, September 28

Linear Combination

${\mbox{Definition: Let }}(u_{i})=(u_{1},u_{2},\ldots ,u_{n}){\mbox{ be a sequence of vectors in }}V$.

${\mbox{A sum of the form:}}{}_{}^{}$

$a_{i}\in F,\sum _{i=1}^{n}a_{i}u_{i}=a_{1}u_{1}+a_{2}u_{2}+\ldots +a_{n}u_{n}$

${\mbox{is called a Linear Combination of the }}u_{i}^{}$.

Span

${\mbox{span}}(u_{i}^{}):=\lbrace {\mbox{ The set of all possible linear combinations of the }}u_{i}^{}\rbrace$

${\mbox{If }}{\mathcal {S}}\subset V\ {\mbox{ is any subset, }}$

${\mbox{span}}({\mathcal {S}}):=\lbrace {\mbox{The set of all linear combination of vectors in }}{\mathcal {S}}\rbrace =\left\lbrace \sum _{i=0}^{n}a_{i}u_{i},\quad a_{i}\in F,u_{i}\in {\mathcal {S}}\right\rbrace$

${\mbox{span}}({\mathcal {S}}){\mbox{ always contains }}0{\mbox{ even if }}{\mathcal {S}}=\emptyset$

Theorem

$\forall {\mathcal {S}}\subset V{\mbox{, span}}({\mathcal {S}}){\mbox{ is a subspace of }}V$

${\mbox{Proof:}}{}_{}^{}$

1. $0\in {\mbox{ span}}({\mathcal {S}})$.
2. ${\mbox{Let }}x\in {\mbox{ span}}({\mathcal {S}})\Rightarrow x=\sum _{i=1}^{n}a_{i}u_{i}{\mbox{, }}u_{i}\in {\mathcal {S}}{\mbox{, }}$

${\mbox{and let }}y\in {\mbox{ span}}({\mathcal {S}})\Rightarrow y=\sum _{i=1}^{m}b_{i}v_{i}{\mbox{, }}v_{i}\in {\mathcal {S}}$

$x+y=\sum _{i=1}^{n}a_{i}u_{i}+\sum _{i=1}^{m}b_{i}v_{i}=\sum _{i=1}^{{\mbox{max}}(m,n)}c_{i}w_{i}$

$\qquad {\mbox{ where }}c_{i}=(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{{\mbox{max}}(m,n)}+b_{{\mbox{max}}(m,n)}){\mbox{ and }}w_{i}\in {\mathcal {S}}$

3.$cx=c\sum _{i=1}^{n}a_{i}u_{i}=\sum _{i=1}^{n}(ca_{i})u_{i}\in {\mbox{ span}}({\mathcal {S}})$

Example 1.

${\mbox{Let }}P_{3}(\mathbb {R} )=\lbrace ax^{3}+bx^{2}+cx+d\rbrace \subset P(\mathbb {R} ){\mbox{, where }}a,b,c,d\in \mathbb {R}$.

${\begin{matrix}u_{1}^{}&=&x^{3}-2x^{2}-5x-3\\u_{2}^{}&=&3x^{3}-5x^{2}-4x-9\\v_{}^{}&=&2x^{3}-2x^{2}+12x-6\end{matrix}}$

${\mbox{Let }}W={\mbox{span}}(u_{1}^{},u_{2}^{}){\mbox{,}}$

${\mbox{Does/Is }}v\in W{\mbox{ ?}}$

$v\in W{\mbox{ if it is a linear combination of span}}(u_{1}^{},u_{2}^{})$

$v=a_{1}u_{1}+a_{2}u_{2}{\mbox{ for some }}a_{1},a_{2}\in \mathbb {R}$

${\mbox{If }}\exists a_{1},a_{2}\in \mathbb {R}$

${\begin{matrix}2x^{3}-2x^{2}+12x-6&=&a_{1}^{}(x^{3}-2x^{2}-5x-3)+a_{2}^{}(3x^{3}-5x^{2}-4x-9)\\\ &=&(a_{1}^{}+3a_{2}^{})x^{3}+(-2a_{1}^{}-5a_{2}^{})x^{2}+(-5a_{1}^{}-4a_{2}^{})x+(-3a_{1}^{}-9a_{2}^{})\end{matrix}}$

${\mbox{Need to solve}}{\begin{cases}2=a_{1}^{}+3a_{2}^{}\\-2=-2a_{1}^{}-5a_{2}^{}\\12=-5a_{1}^{}-4a_{2}^{}\\-6=-3a_{1}^{}-9a_{2}^{}\end{cases}}$

${\mbox{Solve the four equations above and we will get }}a_{1}^{}=-4{\mbox{ and }}a_{2}^{}=2$

${\mbox{Check if }}a_{1}^{}=-4{\mbox{ and }}a_{2}^{}=2{\mbox{ hold for all the 4 equations.}}$

${\mbox{Since it holds, }}v\in W$