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!
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.