Is an unspecified matrix invertible?

[Chillguy]

Homework Statement

Let T be the unique linear operator on C³ for which $$T_{\epsilon1}=(1,0,i), T_{\epsilon2}=(0,1,1), T_{\epsilon3}=(i,1,0).$$
Is T invertible?
2. Homework Equations
If we show T is non singular or T is onto, then this would imply T is invertible.

The Attempt at a Solution

I don't really know where to start, I thought about trying to brute force solve the matrix T but I am quite sure there is a more elegant way and hoping someone can give me a kick in that direction.

 

[Stephen Tashi]

Assuming the scalars for this vector space are the complex numbers, I think $Te_1$ is a linear combination of $Te_2$ and $Te_3$. If that's correct then you can show that T maps a 3-D space to a 2-D space.