I came across the following problem:
Let ${v_{1},v_{2},v_{3},v_{4}}$ be a basis of $\Bbb R^{4}$ and $v=a_{1}v_{1}+a_{2}v_{2}+a_{3}v_{3}+a_{4}v_{4}$ where $a_{i}\in \Bbb R,i=1,2,3,4.$ Then ${v_{1}-v,v_{2}-v,v_{3}-v,v_{4}-v}$ is a basis of $\Bbb R^4$ if and only if
(a) $a_{1}=a_{2}=a_{3}=a_{4},$
(b) $a_{1}a_{2}a_{3}a_{4}=-1,$
(c) $a_{1}+a_{2}+a_{3}+a_{4}\neq1,$
(d) $a_{1}+a_{2}+a_{3}+a_{4}\neq0.$
My attempts: Since ${v_{1}-v,v_{2}-v,v_{3}-v,v_{4}-v}$ is a basis of $\Bbb R^4$ we can express $v=a_{1}v_{1}+a_{2}v_{2}+a_{3}v_{3}+a_{4}v_{4}=a_{1}(v_{1}-v)+a_{2}(v_{2}-v)+a_{3}(v_{3}-v)+a_{4}(v_{4}-v)$ (here, I am not sure whether same scalars ${a_{1},a_{2},a_{3},a_{4}}$ can be used) and hence calculating we get $v( a_{1}+a_{2}+a_{3}+a_{4})=0$ and so $a_{1}+a_{2}+a_{3}+a_{4}=0 $ for $v\neq0.$ Am I right? If not, where did I go wrong? Please help. Thanks in advance for your time.