 Solution

ID: past-exam-of-the-algebra-course-of-the-university-of-oxford/2011/1/b/i/solution

Use the usual unital convention for an algebra over a field. For

a, b, x \in A

and

c \in K

ϕ_{a + c b} (x) = ϕ_{a} (x) + c ϕ_{b} (x), ϕ_{ab} (x) = a (b x) = (ϕ_{a} \circ ϕ_{b}) (x), ϕ_{1} (x) = x .

Passing to matrices in one fixed basis preserves these identities:

M_{a + c b} = M_{a} + c M_{b}

M_{ab} = M_{a} M_{b}

and

M_{1} = I

. Thus

ψ

M_{a} = 0

, evaluating

ϕ_{a}

1_{A}

gives

a = 0

. Its kernel is zero, so the left regular representation is faithful.

 New to topics? Read the docs here!