Example of a linear map $T\epsilon \mathcal{L}(\mathbb{R}^4 \to \mathbb{R}^4)$ such that $\text{dim}[\text{null}(T)] = \text{dim}[\text{range}(T)]$ Give an example of a linear map $T\epsilon \mathcal{L}(\mathbb{R}^4 \to \mathbb{R}^4)$ such that $\text{dim}[\text{null}(T)] = \text{dim}[\text{range}(T)]$.
I came up with $$\text{Let}\ (e_1, e_2, e_3, e_4)\ \text{be a basis of}\ \mathbb{R}^4\ \text{and let}\ T \epsilon \mathcal{L}(\mathbb{R}^4 \to \mathbb{R}^4)\ \text{such that}$$ $$T(e_1)=T(e_2)=0$$$$T(e_3)=e_3$$$$T(e_4)=e_4$$.
Is this example so trivial as to be invalid, and if so, what would have been a better example?

$\textbf{Answer to originally phrased question.}$
You are very close. You can fix it with:
$$T(e_3) = T(e_4) = 0$$
$$T(e_1) = e_3 $$
$$T(e_2) = e_4 $$
Now you have the $\text{null}(T) = \text{span}(e_3,e_4) = \text{range}(T).$
$\textbf{Answer to editted question.}$
Your example is good :)
Any $4 \times 4$ matrix with two columns equal to zero and the other two linearly independent will give you a transformation with this property.
