To clarify: I use $\chi_V$ for the character of the finite representation $V$ of $A$ which is an algebra on a generic field $K$. Then you can consider the idel $[A,A]$ generated by commutators, which is the smallest ideal that makes the quotient $A/[A,A]$ an abelian algebra on $K$. I use $K_C[A]$ for the dual space of the quotient $A/[A,A]$, called space of central functions.
I was watching the proof of a theorem who states: "Let $A$ be a finite and semisimple algebra on a generic field $K$, then $\{ \chi_V | V\in Irr(A) \}$ is a base of $K_C[A]$".
The demonstration i have of this theorem uses two facts which I don't totally understand, in the last part of the proof: $$(a) \space [M_d(K),M_d(K)]=\{X\in M_d(K)|tr(X)=0\};\\(b) \space M_d(K)=[M_d(K),M_d(K)]\oplus K \cdot 1 \space ;$$ where $M_d(K)$ is the algebra of the matrices of order $d\times d$ on the field $K$, $1$ is its unit and $tr(X)$ is the trace of the matrix $X$.
I think to demonstrate those facts I can take a specific base of $M_d(K)$ such as $\{E_{ij},E_{ii}-E_{i+1i+1},1|i\ne j, i=1,...,d-1\}$, which as $d^2-1$ elements with trace equal to zero, and do the math, but I don't see it.
If someone can help me I'll much appreciate it!
Edit: as Captain Lama reminded me in the comments, the proof was done under the hypothesis that $char(K)=0$ for semplicity, but I think this two facts, at least, should be provable with the weaker hypothesis that $d$ doesn't divide the characteristic of $K$.