How to prove stuff about linear algebra?

[*melinda*]

How to prove stuff about linear algebra?

Question:

Suppose (v_1, v_2, ..., v_n) is linearly independent in V and w\in V.
Prove that if (v_1 +w, v_2 +w, ..., v_n +w) is linearly dependent, then w\in span(v_1, ...,v_n).

To prove this I tried...

If (v_1, v_2, ..., v_n) is linearly independent then a_1 v_1 + ...+a_n v_n =0 for all (a_1 , ..., a_n )=0.
then,
a_1 (v_1 +w)+a_2 (v_2 +w)+...+a_n (v_n +w)=0
is not linearly independent, but can be rewritten as,
a_1 v_1 + ...+a_n v_n +(\sum a_i )w=0
so,
a_1 v_1 + ...+a_n v_n = -(\sum a_i )w.
Since w is a linear combination of vectors in V, w\in span(V).

Did I do this right?
Is there a better way of doing this?
Any input is much appreciated!

 

[LeonhardEuler]

Your proof is pretty much correct, but in this sentence:

If (v_1, v_2, ..., v_n) is linearly independent then a_1 v_1 + ...+a_n v_n =0 for all (a_1 , ..., a_n )=0.

I think you mean to say:
If (v_1, v_2, ..., v_n) is linearly independent then a_1 v_1 + ...+a_n v_n =0 only when each a_i=0

 

[*melinda*]

Yes, that would make a bit more sense. Sometimes I understand what I mean to do, but don't know how to say it.

Thanks a bunch!

 

[Hurkyl]

If (v_1, v_2, ..., v_n) is linearly independent then a_1 v_1 + ...+a_n v_n =0 for all (a_1 , ..., a_n )=0.

This is wrong. If the collection of vectors is independent, and if a_1 v_1 + ...+a_n v_n =0 then a_1 = a_2 = \cdots = 0.