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As the title says, what can we say about $v_1+w_1, v_2+w_2, ..., v_n+w_n$ if $v_1, v_2, ..., v_n$ and $w_1, w_2, ..., w_n$ are linearly independent vectors of $R^n$?

I've tried approaching this problem by observing that $dimV=dimW=n$, where $V=[v_1, v_2, ..., v_n]$ and $W=[w_1, w_2, ..., w_n]$. Then $v_1+w_1, v_2+w_2, ..., v_n+w_n$ are linearly independent if and only if $dim(V+W)=n$. It follows that $dim(V \cap W)=n$, but then I get stuck.

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    $\begingroup$ You can't say anything about it. Suppose $w_i=-v_i$... $\endgroup$ Feb 12 at 19:27
  • $\begingroup$ Re prev comment - versus $w_i = v_i.$ $\endgroup$ Feb 12 at 20:50
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Hint.

If $v_k$ are LI then they form a basis for $\mathbb{R}^n$ and then

$$ w_k = \sum_j a_j^k w_j $$

or

$$ W = A V $$

Now $V+W = (A+I_n)V$ so if $A+I_n$ is rank $n$ then $V+W$ are a LI set. Note that it is well possible that $\text{rank}(A+I_n)< n$ As an example take $A = -I_n$

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