is a matrix and is a linear transformation

1)Show that with either has no solution or has infinitely many

2)Show that with has rank always has a solution

3)Show that with has rank has at most one solution

4)Show that where and has rank has precisely one solution

I'm not sure how to start on either method with 1)

For 2) doing the RRE form method, can be reduced down to a matrix with no nonzero rows, but I get stuck there.

I'm confused about 3, if has rank does that mean that ?

4) I understand the RRE form method because it can be reduced to the identity matrix, but I don't know how to prove it using the rank nullity formula

Any help would be very much appreciated!