Example of a linear transformation where kernel = image

Question

Let $T$ be the linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Find an example of such $T$ where $\ker T = \text{im } T$.

Answer 1

Consider the linear map defined by $T(1,0) := (0,0)$ and $T(0,1) = (1,0)$. In matrix form,

$$ \begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}.$$