1. ## Linear Independence

Let V be a vector space over field F.Let V* be dual of V.If {f1,f2,...fk}subset of V* and {v1,v2,...vk}subset of V such that fi(vj)= 1 ,if i=j and 0 ,if i not equals j ..then prove that {f1,...fk} is linearly independent in V* and {v1,...vk} is linearly independent in V..

i can show that {f1,...fk} is Linearly independent..i need help as to how to show {v1,...vk}are linearly independent in V..

2. Originally Posted by math.dj
Let V be a vector space over field F.Let V* be dual of V.If {f1,f2,...fk}subset of V* and {v1,v2,...vk}subset of V such that fi(vj)= 1 ,if i=j and 0 ,if i not equals j ..then prove that {f1,...fk} is linearly independent in V* and {v1,...vk} is linearly independent in V..

i can show that {f1,...fk} is Linearly independent..i need help as to how to show {v1,...vk}are linearly independent in V..

Suppose $\displaystyle \sum\limits_{i=1}^ka_iv_i=0\,,\,\,a_i\in \mathbb{F} \Longrightarrow\,\forall\,1\leq j\leq k\,,\,\, 0=f_j(0)$ $\displaystyle =f_j\left(\sum\limits_{i=1}^ka_iv_i\right)=\sum\li mits_{i=1}^kf_j(a_iv_i)=\sum\limits_{i=1}^ka_if_j( v_i)=a_j$ $\displaystyle \Longrightarrow\,a_j=0\,\,\forall j$ and we're done.