IMC 1997 · Problem 4

a) Let the mapping $f : M_n \to \mathbb{R}$ from the space $M_n = \mathbb{R}^{n^2}$ of $n \times n$ matrices with real entries to reals be linear, i.e.:

$$f(A + B) = f(A) + f(B), \quad f(cA) = cf(A) \tag{1}$$

for any $A, B \in M_n$, $c \in \mathbb{R}$. Prove that there exists a unique matrix $C \in M_n$ such that $f(A) = \operatorname{tr}(AC)$ for any $A \in M_n$. (If $A = \{a_{ij}\}_{i,j=1}^{n}$ then $\operatorname{tr}(A) = \sum\limits_{i=1}^{n} a_{ii}$).

b) Suppose in addition to (1) that

$$f(A.B) = f(B.A) \tag{2}$$

for any $A, B \in M_n$. Prove that there exists $\lambda \in \mathbb{R}$ such that $f(A) = \lambda . \operatorname{tr}(A)$.