It's probably better to start with the RHS and get the LHS.

ie.

It's also clear that

or

Take a basis vector of

, call it

. Do the same for

, call the basis vector

.

I have to get

We get:

since

and i'm now working on a vector in

. The linear map

also has to act on a vector, so it's acting on some fixed

.

Here

but linear maps are defined on how they work on a vector so I need to put a

in there.

Apply

to both sides to get:

From here i'm absolutely lost! Does anyone have any ideas??

(P.S This is very hard!)