Dual space

• Nov 7th 2009, 09:07 PM
math.dj
Dual space
Let V be a vector space over field F.Let g,f1,f2,...fk belong to V* i.e;dual of v,then show that g belongs to span{f1,f2,...,fk} if and only if intersection of kernel(fi) i=1 to k is subset of Kernel(g)..

• Nov 8th 2009, 06:32 AM
tonio
Quote:

Originally Posted by math.dj
Let V be a vector space over field F.Let g,f1,f2,...fk belong to V* i.e;dual of v,then show that g belongs to span{f1,f2,...,fk} if and only if intersection of kernel(fi) i=1 to k is subset of Kernel(g)..

In any vector space we have that $x\in Span\{v_1,...,v_n\}\Longleftrightarrow x \mbox{ is a linear combination of } v_1,...,v_n$.
We can assume $f_1,...,f_k$ are lin. indep.
Thus in our case: $g\in Span\{f_1,...,f_k\}\Longrightarrow g=\sum\limits_{i=1}^ka_if_k$ $\Longrightarrow \,\forall\,x\in\bigcap\limits_{i=1}^kKer(f_k)\,,\, \;g(x)=\sum\limits_{i=1}^ka_if_i(x)=0\Longrightarr ow\,x\in Ker(g)$ $\Longrightarrow\bigcap\limits_{i=1}^kKer(f_i)\subs et Ker(g)$
OTOH, if $g\notin Span\{f_1,...,f_k\}$ then $\{f_1,...,f_k,g\}$ is lin. indep, so we can complete it to a basis $X=\{f_1,...,f_k,g,h_1,...,h_r\}$ of $V^{*}$, and let $\{v_1,...v_k,v_g,u_1,..,u_r\}$ be the dual basis of $X$ $\Longrightarrow\, \forall\,1\leq i\leq k\,,\,\,f_i(v_g)=0\,\,but\,\,g(v_g)=1\Longrightarr ow \bigcap\limits_{i=1}^kKer(f_i)\nsubseteq Ker(g)$ and we're done.