Question 455735: If V={v1,v2,...,vk} is linearly independent, and w is not element of V. Then show that {v1+w,v2+w,...,vk+w} is linearly independent
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Let  +...+ .
Now let
 +...+ =  ,
which for the purpose of contradiction, we assume that not all the a's are 0, (i.e., the set is linearly DEPENDENT).
==> ( +...+ )*
+( +...+)* + ...+( +...+ )* =
By the hypothesis, {v1, v2, v3,..., vk } is linearly independent, and thus,
+...+
+...+
...................................
+...+ .
We get a homogeneous system of equations.
Adding all corresponding sides of the system, we get
 +...+
OR,
+...+.
Subtracting this equation from each one of the equations in the system above, we obtain
 =...=,
CONTRARY to the initial assumption that not all a's are 0.
Hence {v1+w,v2+w,...,vk+w} must be linearly independent.
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