I am self-studying Linear Algebra from Paul Halmos'"Finite Dimensional Vector Spaces" and I am struggling with the following exercise which is found just before section 8.
The first part of the question is to find a necessary and sufficient condition for linear dependence of two vectors in $\mathbb{C}^3$ which I think I managed to do:
Two vectors $\vec{x}=(x_1,x_2,x_3)$ and $\vec{y}=(y_1,y_2,y_3)$ in $\mathbb{C^3}$ are linearly dependent if and only if $x_1y_2=x_2y_1,\ x_1y_3=x_3y_1$, and $x_2y_3=x_3y_2$.
I am struggling with the second part of the question:
Find a similiar neccesary and sufficient condition for linear dependance of three vectors in $\mathbb{C^3}$.
I have tried assuming that the vectors are linearly dependent and making some assumptions to try and guess what a necessary and sufficient condition might be and then prove that it is indeed one which is how I solved the first part of the question. To be more concrete I guessed the condition for the first part of the question like this:
If $\vec{x}$ and $\vec{y}$ are linearly dependent then for suitable scalars $\alpha,\beta \in \mathbb{C}$ not all zero we have $$ \begin{array}{c} \alpha x_1+\beta y_1=0\\ \alpha x_2+\beta y_2=0\\ \alpha x_3+\beta y_3=0 \end{array} $$ If $\alpha$ and $\beta$ are both not zero then we can conclude that $x_1y_2=x_2y_1,\ x_1y_3=x_3y_1$, and $x_2y_3=x_3y_2$.
I tried adopting a similiar strategy to guess the condition for the case of three vectors in $\mathbb{C^3}$ and I also tried, before writing this up, limiting myself to the case of $\mathbb{R^3}$ and using the fact that if the scalar triple product of three vectors is equal to zero then they are co-planar to try and see if I could get similar condition like in part one of the question, but I have yet to make any progress.
I am still racking my brain but I would like to ask for any help with this problem.
EDIT: Thank you Maximilian Janisch and WE Tutorial School for the hints you have left in the comments. If anyone has any other way of attacking this problem without the use of determinants I would love to hear about it as well.