Past exam of the algebra course of the University of Oxford / 2011 / 1 / b / i

Use the usual unital convention for an algebra over a field. For $a,b,x\in A$ and $c\in K$,Passing to matrices in one fixed basis preserves these identities: $M_{a+cb}=M_a+c{M}_b$, $M_{ab}=M_a{M}_b$ and $M_1=I$. Thus $\psi$ is an algebra homomorphism. If $M_a=0$, evaluating ${\varphi }_a$ at $1_A$ gives $a=0$. Its kernel is zero, so the left regular representation is faithful.