Write $M_n(F)$ for the space of $n\times n$ matrices with entries in $F$. Let $sl_n(F)=\{A\in M_n(F)|Tr(A)=0\}$ and $pgl_n(F)=M_n(F)/F\cdot I_n$. Show that the quotient map restrict to an isomorphism $sl_n(F)\rightarrow pgl_n(F)$.
My attempt:
The quotient map $\pi:M_n(F)\rightarrow pgl_n(F)=M_n(F)/F\cdot I_n$ is defined by $A\rightarrow F\cdot I_n+A$. To prove that it restrict to an isomorphism on $sl_n(F)$, is equivalent to proving it's linear and bijective. It's linear because $\pi(\alpha A+A')=F\cdot I_n +(\alpha A+A')=(F\cdot I_n+\alpha A)+(F\cdot I_n+A')=\alpha (F\cdot I_n+A)+(F\cdot I_n+A')=\alpha\pi(A)+\pi(A')$. Remains to show it is bijective i.e. injective ad surjective. It's injective because suppose it's not, then for some $\lambda_1\ne\lambda_2$, $A'\ne A$: $I_n\lambda_A+A'=I_n\lambda_2+A\implies n\lambda_1=n\lambda_2\implies\lambda_1=\lambda_2$ a contradiction. It's surjective by definition. Hence the quotient map restrict to an isomorphism.