# If $v_1, v_2, ..., v_n$ are linearly independent and so are $w_1, w_2, ..., w_n$, what can we say about $v_1+w_1, v_2+w_2, ..., v_n+w_n$?

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.

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

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 $$A = -I_n$$

• This is a brilliant answer and exactly what I was looking for! Feb 12 at 20:06