linearly independent and dependent

Problem 1:

If are linearly independent vectors in , show that is linearly independent set.

Proof 1:

If where , since is linearly independent.

Is the proof correct? I felt something missing in the proof.

Problem 2:

If are linearly dependent vectors in , show that is linearly dependent set for any .

Proof 2:

If . Since are linearly dependent, for some . Thus, is linearly dependent.

Please check my proofs.

Problem 3:

Let and , is invertible .

Show that are linearly independent if and only if are linearly independent vectors in .

How to prove problem 3? what theorem do I need?