Okay, it looks like I had the right idea, but was wrong in my process.

By your definition of the linear transformation, given a vector

the linear transformation

is given by

.

So we are transforming a vector in

to a scalar in

. So what does this transform

look like? First, we must realize that we are multiplying on the left. Thus,

must be

since

is

and we need a scalar.

Now remember in my previous post that when we multiply vectors and matrices we get linear combinations. In this case our matrix is actually a vector and the linear combination is

. Thus,

is given by

.

Now you can see why I delete my previous post, it was wrong.