Proving a Unique Linear Transformation

Question

Let $n\in \mathbb{N}$, and let tr: $\mathbb{R}^{n\times n}\rightarrow \mathbb{R}^{n\times n}$ be the trace map. Recall that a matrix $P\in \mathbb{R}^{n\times n}$ is a projection if $P^2 = P$

Prove that tr is the unique linear transformation $T:\mathbb{R}^{n\times n} \rightarrow \mathbb{R}$ suchthat $T(P) = rank(P)$ for every projection $P \in \mathbb{R}^{n\times n}$.

What does it mean to prove the linear transformation is unique? I proved in the previous part to this problem that for every projection $tr(P) = rank(P)$. How is the question any different?

Answer 1

It means that you need to show if some other linear transformation $T:\Bbb R^{n\times n}\to\Bbb R$ satisfies that for every projection $P$, $T(P)=\text{rank}(P)$, then as functions, $T=\text{tr}$.

Answer 2

Well, in the previous part you prove that trace is one such linear transformation. Now you need to prove there cannot be another such, i.e. one that is different from the trace map...

As to how to do it, the first thing that comes to mind is that a linear transformation is determined by its effect on a basis...