Null space and range

Let's denote the set of $2 \times 2$ matrices by $M_{22}$.

The trace function $\mathrm{tr}:\, M_{22} \rightarrow \mathbb{R}$ is given by:

$$\mathrm{tr}(A) = a_{11} + a_{22}$$Compute the dimension of the null space of $\mathrm{tr}$.

Hint: You can calculate the null space directly or just use the dimension theorem.

Hint: The dimension of $M_{22}$ is $4$, while that of $\mathbb{R}$ is $1$.