A. Now there are two conventions for the definition of the matrix for T. The convention I am used to the vectors are column vectors, and we write the action as Tx, the other treats the vectors as row vectors and has action xT.

In the first case the columns are the images of the natural basis of the domain

In the second case the rows are the images of the natural basis:

B. You check this

C. The last component of for all is zero so is of this form?

D. Look at the answer to C. to see if is onto. To see if it is one-to-one consider if there are distinct and such that:

RonL