Calculating a parameter for which three complex vectors are linearly independent

Question

Given are the vectors $$(1, 2 + i, 0), (1-i, 2, 2), (a, 2i + 5, 1) \in \mathbb{C}$$

I`m suppossed to figure out for which values of $a\in \mathbb{C}$ are these vectors linearly independent.

I tried figuring out the solution by solving the linear system of equations, but quickly figured out that this isn't getting me anywhere.

By trying out some values I figured out that for $\lambda_1 = -2, \lambda_2 = -\frac{1}{2}, \lambda_3 = 1$ the vectores are linearly dependent for $\alpha = \frac{1}{2}(5-i)$. This isn't a good approach to this problem though, and I'm kind of stuck here and don't know how to analytically find values for $a$. Can anybody help me out?

Thanks!

Answer 1

Hint

Let us write the vectors as

$$U_1=(1,2,0)+i(0,1,0)$$

$$U_2=(1,2,2)+i(-1,0,0)$$

$$U_3=(a_1,5,1)+i(a_2,2,0)$$

from here we see that it should be

$$U_3-2U_1=\lambda U_2$$

$$\implies \lambda =\frac{1}{2}$$

$$\implies a_1-2=\frac{1}{2}, a_2=-\frac{1}{2}$$

$$\implies a=\frac{1}{2}(5-i)$$

Answer 2

Since the first two vectors are linearly independent, we need only find the values of $a$ for which the the last vector lies in the span of the first two.

Let the three vectors be $v_1,v_2,v_3$ and suppose $v_3 = x v_1 + y v_2$. If you look at the last two components, then there are unique $x,y$ such that the bottom two components match. Then if the three vectors are dependent, we must have $a = x [v_1]_1 + y [v_2]_1$.