Prove that vectorsv1, v2, ... ,vn of the same dimension are linearly dependent if and only if one of them can be written as a linear combination of the others.
I'm not sure how exactly I would correctly go about proving this. Thanks!
Well, this can be proven directly:
Theorem: of the same dimension are linearly dependent iff one of them can be written as a linear combination of the others.
Proof:
Let the set be linearly dependent. This means that there exists scalars , not all zero, such that:
So, assume that . We can rewrite this as:
Therefore, we get as a linear combination of the others.
Now conversely, we assume that is a linear combination of which means that there exists scalars such that:
We can rewrite this to be:
Therefore, the set is linearly dependent.