I think I've got a rather glaring hole in my understanding of this material. I've read and reread the text, watched the MIT Open Courseware lectures, and beat my head against this material for quite a while now. Any help would be much appreciated. Without further ado...

1) Define a linear transformation by . Consider ordered bases and for and respectively.

a) Find the representation matrix of with respect to the given bases and .

b) Express in terms of

2) Let be the vector space of all 2 x 2 matrices (over . Consider

, , , ; and .

It is known that is an ordered basis for . Define by for all = .

Edit: Couldn't get the format right for that x matrix. Is should be x_{11}, x_{12}, x_{21}, x_{22}.

a) Find , the coordinate vector of with respect to the ordered basis .

b) Is L a linear transformation?

c) Find the representation matrix of L with respect to , if L is a linear transformation.

For 1), I found . Then I tried to set up a matrix with the bases of C and those values. I interpreted the bases of C as the standard bases (1, 0), (0, 1). I really don't think that's right, so I'm kind of stuck there.

For 2), I can't even make headway on the first part. I set up the problem like I would to find a coordinate vector, but it comes out as a 2 x 10 matrix. Trying to account for the zero columns still leaves me with a 2 x 6. I believe that this is a linear transformation, but as I said, I'm quite stumped.

Any help would be very much appreciated.