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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$span (v_1,...,v_m) = m.$

Can someone please help me? I am stuck. Any help would be really appreciated. Thanks!

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marked as duplicate by Arnaud D., Jyrki Lahtonen, Adrian Keister, ancientmathematician, Dennis Gulko Sep 3 '18 at 16:35

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  • $\begingroup$ Your first line in your attempt seems to have something backwards, independent mean there are no such constants (other than 0,0,0,0,0,0...) Also I have no idea how you are getting this spanning conclusion or the independence, as it's not at all clear how it follows from your givens. Consider expanding some details there. $\endgroup$ – Adam Hughes Sep 21 '16 at 21:23
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    $\begingroup$ 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. $\endgroup$ – user4242 Sep 21 '16 at 21:30
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Hint:

Show the span contains $v_1-v_2$, $\dots,v_1-v_m$, and show these vectors are linearly independent.

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