Hi I would try this way, just briefly:
If are linearly dependent, then there exist scalars such that
Denote the matrix, to which determinant belongs, its j-th row and .
Then there holds , since for each there holds
so nonzero vector belongs to and thus is singular matrix.
The other way,
if is singular matrix, there exists nonzero vector such that for all there holds
, where . But then so
and are linearly dependent