Dual space

**math.dj** · November 7th 2009, 09:07 PM

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)..

Thank you in advance..

**tonio** · November 8th 2009, 06:32 AM

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)..

Thank you in advance..

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.

Tonio

**math.dj** · November 10th 2009, 05:14 AM

Thank a ton..

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