Answer:
See the proof below.
Step-by-step explanation:
What we need to proof is this: "Assuming X a vector space over a scalar field C. Let X= {x1,x2,....,xn} a set of vectors in X, where
. If the set X is linearly dependent if and only if at least one of the vectors in X can be written as a linear combination of the other vectors"
Proof
Since we have a if and only if w need to proof the statement on the two possible ways.
If X is linearly dependent, then a vector is a linear combination
We suppose the set
is linearly dependent, so then by definition we have scalars
in C such that:

And not all the scalars
are equal to 0.
Since at least one constant is non zero we can assume for example that
, and we have this:

We can divide by c1 since we assume that
and we have this:

And as we can see the vector
can be written a a linear combination of the remaining vectors
. We select v1 but we can select any vector and we get the same result.
If a vector is a linear combination, then X is linearly dependent
We assume on this case that X is a linear combination of the remaining vectors, as on the last part we can assume that we select
and we have this:

For scalars defined
in C. So then we have this:

So then we can conclude that the set X is linearly dependent.
And that complet the proof for this case.