What I am unable to understand is why we can "we can associate each polynomial to a vector by taking it's leading coefficients" and then place them in a matrix to compute the determinant. This is in reference to: Answer
When we are talking about plain old vectors in $R^3$ for instance, it seems natural but what is it about the coefficients of a polynomial? Would we also be able to reduce this matrix to Echelon form to find a set of linearly independent polynomials from the given set? That is to say, the nonzero rows would be linearly independent.