Need help with this proof

Question

Let {~v1, · · · ~vm} ⊂ Rn be a linearly independent set. Let L ∈ Rm×n . Prove that, if null(L) = {~0}, then {L~v1, · · · L~vm} ⊂ Rm is a linearly independent set.

Answer 1

Proof:

Suppose c_1 L(v_1) + c_2 L(v_2) + ... + c_m L(v_m) = 0, where c_i ∈ R.

(We need to show that c_1 = c_2 = ... = c_m = 0)



Since L is linear, it follows that

L(c_1 v_1 + c_2 v_2 + ... + c_m v_m) = 0

But Null (A) = 0, so

c_1 v_1 + c_2 v_2 + ... + c_m v_m = 0.

Now, since {v_1, v_2, ..., v_m} is a linearly independent set, we have

c_1 = c_2 = ... = c_m = 0,

Therefore, {L(v_1), L(v_2), ..., L(v_m_)} is a linearly independent set. ◊