Given the linear transformation:

$T_3 : \mathbb{C}^3 \to \mathbb{C}^3 , (x_1, x_2, x_3) \mapsto (x_1+x_2, x_1+x_3, x_1 - x_2+2x_3)$

I need to find the image and kernel, and basis for both of them. I know that the kernel is the set of vectors for which the linear transformation will map to zero, and this can be found by setting up a system of equations like such:

$x_1 + x_2 = x_1 + x_3 = x_1 - x_2 +2x_3 = 0$

But once here, how would I format my answer for $ker(T_3)$, and how would I then find a basis for that? Secondly I understand the image of a linear transformation to be the span of the target vectors from the linear transformation, so would the $im(T_3)$ just be $\mathbb{C}^3$ ?

Thank you!