# Prove that dim[(span$(v_1 + w,…,v_m + w)] \geq m - 1.$ [duplicate]

Suppose $v_1,...,v_m$ is linearly independent in $V$ and $w \in V.$ Prove that dim[(span$(v_1 + w,...,v_m + w)] \geq m - 1.$

attempt: Let $v_1,...,v_m$ be linear independent. So there is $a_1,..,a_m \in F,$ such that $a_1v_1 + .... + a_mv_m = 0$. where $a_1=...=a_m = 0.$

Then $(v_1 - v_2,...,v_{m-1} - v_m)$ span and are linear independent in $V$. And dim$span (v_1,...,v_m) = m.$

• I dont really know how to continue. I did an exercise before and showed $v_1 - v_2,...,v_{m-1} - v_m$ was linearly independent and that it spanned a vector space V , whenever, $v_1,...,v_m$ are linearly independent. So i thought of using it. – user4242 Sep 21 '16 at 21:30
Show the span contains $v_1-v_2$, $\dots,v_1-v_m$, and show these vectors are linearly independent.