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.