let v1,v2,....,vn be linearly independent vectors in a vector space V . show that v2,v3.........,vn cannot span V ?

Question

let v1,v2,....,vn  be linearly independent vectors in a vector space V . show that v2,v3.........,vn cannot span V ?

anonymous · Answer

$\{v_1,v_2,\cdots,v_n\}$ are linearly independent means that $c_1v_1+c_2v_2+\dots+c_nv_n=0$ only for $c_i=0$. if $v_2,v_3\dots,v_n$ span $V$ then we can surely express $v_1$ in terms of the others, so: $$v_1=k_1v_2+k_2v_3+\dots+k_{n-1}v_n$$but this suggests that we have $$v_1-k_1v_2-k_2v_3-\dots-k_{n-1}v_n=0$$which implies $v_1,v_2,\dots,v_n$ are linearly dependent -- contradiction

anonymous · Answer

since the bottom line suggests we could pick $c_1=1,c_2=-k_1,c_3=-k_2,\dots,c_n=-k_{n-1}$ to satisfy $c_1v_1+c_2v_2+\dots+c_nv_n=0$