(1) be the n-th order homogeneous differential equation which is represented in matrix form (2)

If are the solutions to (1), what are the correspondence solutions of (2)?

are the solutions to (1), then the corresponding solutions to 2 are the vectors for all i.

Is this the correct matrix for solutions

Prove that the following two conditions are equivalent: Please help me get started.

(i) are linearly independent;

(ii) not equal to zero for some in the interval I.

Thank you.