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.

Please check you haven't yet proved anything... . You could argue as follows: Suppose since we're given is lin. ind.
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.

This is similar as the above one but at least here you mentioned that the given set of vectors is l.i....I'd wrap it up as follows: Since is l.d. we have a trivial linear combination and for some and STILL for some index is l.d.
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?