Going through some old exams and found this problem that i cant seem to get my head around and i couldnt find any solutions for it either.
Suppose , n≥m, so that $\overrightarrow{v_{1}},......,\overrightarrow{v_{m}}$ is linearly independent in $\mathbb{R}^n$ and $\overrightarrow{w}$∈ $\mathbb{R}^n$. And we also suppose that $\overrightarrow{v_{1}}-\overrightarrow{w},......,\overrightarrow{v_{m}}-\overrightarrow{w} $ is linearly dependant.
Show that $\overrightarrow{w}$ ∈ span$(\overrightarrow{v_{1}},......,\overrightarrow{v_{m}})$
If $w = 0$ we are done. Suppose $w \not= 0$
There exist some $a_{1},a_{2},\cdots, a_{n}$ not all null such that $\sum_{i=1}^{n}(v_{i} -w)a_{i} = 0$
Then we have that $\sum_{i = 1}^{n} v_{i}a_{i} = w\sum_{i = 1}^{n}a_{i}$
if $\sum_{i=1}^{n}a_{i} = 0\implies \sum_{i = 1}^{n} v_{i}a_{i} = 0 \implies$ all $a_{i} = 0$ because $\{v_{1},\cdots,v_{n}\}$ are linearly independent, this is a contradiction because we had that they were not all null.
Then $\sum_{i=1}^{n} a_{i} = a \not= 0$ and thus $\sum_{i = 1}^{n} v_{i}(a_{i}a^{-1}) = w$
Thus $w \in $ span$(v_{1},\cdots,v_{n})$