Linear Independence and Intersections of Sets

[TranscendArcu]

Homework Statement

Let E' and E'' be linearly independent sets of vectors in V. Show that E' \cap E'' is linearly independent.

The Attempt at a Solution

To show a contradiction, let E' \cap E'' be linearly dependent. Also let A be all of the vectors in E' \cap E''. Thus, A \subseteq E' and A \subseteq E''. Because A is linearly dependent, there exists A_1,...,A_n distinct vectors in A such that

a_1 A_1 + ... + a_n A_n = \vec0, where a_1,...,a_n are not all zero. But if such a nontrivial linear combination of vectors in A exists, then E' must be linearly dependent since A \subseteq E'. But this is contrary to our definition that E' is linearly independent. This is similarly contradictory for E''. Thus, it is shown that E' \cap E'' cannot be linearly dependent and must rather be linearly independent.

First of all, I don't know if this proof is correct (although it seems conceivable to me). Also, I didn't know how to prove the problem statement directly, so I had to do it by contradiction. If anyone could give me a hint as to how to do this directly, I would be grateful.

 

[TachyonRunner]

Your proof seems to do the job, though I recommend you consider the case when E'\cap E'' is empty separately first.

If you have some theorems at your disposal then you can shorten up your proof considerably by simply noting that a non-empty subset of a linearly independent set is itself linearly independent.