the identity matrix, and two invertible matrices.
a) What about ?
The first equality is true because is a ring.
c) What about and associativity of multiplication.
d) , so
e)Try with and
f) What we can write in any ring is
You get the asked equality iff , that is to say iff . So e) answers the question.
2) You just have to check how many vectors are independant in each set. Note that a subspace of has a maximal dimension of , so if a set of vectors contains at least independant vectors, then