IFF statement reworded, which is equivalent?

Question

The problem is given as:

Prove that a subset {u₁, . . ,u_p} in V is linearly independent if and only if the set of coordinate vectors {[u₁]_B, . . , [u_p]_B} is linearly independent in ℝⁿ.

The problem was reworded as (note it's no longer an IFF):

Prove if {u₁, . . ,u_p} in V is linearly independent, then {[u₁]_B, . . , [u_p]_B} is linearly independent in ℝⁿ.

I'm having a hard time seeing if the reworded problem is equivalent to the original problem going forwards or backwards, since the original problem is an IFF.

If I were to prove the reworded problem by contrapositive by supposing one of the sets are linearly dependent, which set do I start with being dependent?

Answer 1

The rewording is one direction of the original biconditional. The original statement implies the rewording but it is not obvious that the new statement implies the original. In this case you can derive the original so they are equivalent in the sense of both being true.

If you want to prove the rewording by using the contrapositive, you assume the set {[u₁]_B, . . , [u_p]_B} is dependent and show the other one is.