We have given by . For , iff so, iff has the form:
with . Easily proved, those three matrices are linearly independent and as a consequence form a basis for .
See attachments for problem and note that I want to solve the problem for n=2 only. Also, my attempted solution is attached as well.
N(T) denotes the nullspace of T. I found the nullspace, but I'm not sure if I have the correct basis for it or if I have written the solution space with the correct notation. Obviously, the nullspace will be the set of vectors where the 2 diagonal components of the matrix are additive inverses of one another. But given that either of the diagonal components could be negative, how do I write this in proper format?
Also, did I get the correct basis?
Thanks for your time.