Prove the following vectors are linearly independent

Question

So I have these three vectors: [i, 2+i, 3]; [2, -i, 4-i}; [3, -1, 2] and I need to show they are linearly independent. This means that given scalars $x_1, x_2, x_3$ their scalar sum should equal 0. How would deal with complex numbers in this case? Would one still perform normal Gaussian elimination or is there some quicker way?

Answer 1

Consider $c_1,c_2,c_3$ and linear combination of the above vectors

$c_1i+2c_2+3c_3=0$

$2c_1+ic_1-ic-2-c_3=0$

$3c_1+4c_2-ic_2+2c_3=0$

Equating like terms we get

$c_1=0,2c_2+3c_3=0$ ;$2c_1-c_3=0,c_1-c_2=0$

from there we get $c_1=c_2=c_3=0$