# How to find the kernel and image of a linear mapping?

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!

The only solution of that system is $$(0,0,0)$$. So, $$\ker(T_3)=\{(0,0,0)\}$$, the only basis of which is $$\emptyset$$. And, by the rank-nullity theorem $$\dim\operatorname{Im}(T_3)=3$$ and therefore $$\operatorname{Im}(T_3)=\mathbb C^3$$.