Linear independent vectors

Question

I have this problem and need your help : $V_1$, $V_2$, $V_3$ are linearly independent vectors.

And $W_1 = 2 V_1 + V_2\ ;\ W_2 = V_1 -2V_2 + V_3\ ;\ W_3 = V_3 -3V_2$

Are those three vectors also linearly independent ? I thought about using the determinant to find out but it is way too long, Does anyone have an easier way ? Thanks a lot.

Answer 1

Another way you can think about it:

Take $c_1, c_2, c_3$ such that $c_1W_1 + c_2W_2 + c_3W_3 = 0.$

Now check whether all $c_i's$ $(i = 1, 2, 3)$ are zero or not. If all $c_i's$ are zero then three vectors are linearly independent otherwise dependent.

Answer 2

Consider the given equalities as a system of linear equations with unknown $v_1, v_2, v_3.$ Then $w_1,w_2,w_3$ is linear independent iff determinant of system is non-zerow.