# The linear independence of vectors $(u_1+iv_1)$ and $(u_2+iv_2)$ in complex vector space $V_C$

Suppose $$V_C$$ is a complex vector space and hence the vectors in it should be in the form of $$(u+iv)$$ where $$u,v \in V$$ and $$V$$ is a real vector space

Supposed there are two vectors $$(u_1+iv_1)$$ and $$(u_2+iv_2)$$ are linearly independent in $$V_C$$. Is there any conclusion about the linear independence or dependence that we can make about the vectors $$u_1,v_1,u_2,v_2\in V$$?

Since $$(u_1+iv_1)$$ and $$(u_2+iv_2)$$ are linearly independent in $$V_C$$, it's clear to state that $$(a_1+ib_1)(u_1+iv_1)+(a_2+ib_2)(u_2+iv_2)=0\implies a_1=b_1=a_2=b_2=0$$ where $$a_1,b_1,a_2,b_2\in \mathbb R$$

This is equivalent to say(that is expand the above equation): $$\begin{cases}a_1u_1-b_1v_1+a_2u_2-b_2v_2=0 & \text{real part} \\ a_1v_1+b_1u_1+a_2v_2+b_2u_2=0 &\text{imaginary part} \end{cases} \iff a_1=b_1=a_2=b_2=0$$

Then I don't know whether this implies that $$(u_1,v_1,u_2,v_2)$$ are linearly independent or other relationships about linear independence and dependence within these four vectors.

Thus any help? Thanks!

No conclusion can be reached about the dependence of $$u_1,u_2,v_1,v_2$$. As a simple example, note that the vectors $$(1+i,1+i), \quad (1+i,1-i)$$ are linearly independent in $$\Bbb C^2$$, but the real vectors $$(1,1),(1,1),(1,1),(1,-1)$$ are clearly linearly dependent over $$\Bbb R^2$$.