Are eigenvectors independent



I know about (and have seen proof of) the fact that eigenvectors
corresponding to distinct eigenvalues form a lin. independent seet.
(**)

But what about this type of situation:

Say a matrix A has 5 eigenvalues (where one of them as multiplicity 2
as a root of characteristic equation of A).

We are able to get 2 independent eigenvectors for this multiplicity 2
eigenvalue.
Is it true that these 2 eigenvectors along with the other 4 (each
related to a multiplicity 1 root) make an independent set? What is the
reasoning behind this?

At first I thought it was obvious (by using theorem (**) 2 times), but
then I realized it wasn't.

Say we have 3 vectors a,b,c. If {a,b} is indep. and {b,c} is indep and
{a,c} is indep., how can we say that {a,b,c} is independent? (I
actually thought this was obviously true, but then realized it wasn't
true for many cases).

.

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