As always, is the ring of matrices with complex entries.

**Problem**. Let

i) Show that is a vector space but is not.

ii) Show that

**Solution**. Let be the standard basis of

i) Let and Then

So and thus is a vector space. To show that is not a vector space, see that, for example, because but because

ii) Since all the eigenvalues of a nilpotent matrix are zero, the trace of a nilpotent matrix zero and thus implying that because is a vector space.

So, to complete the proof of ii), we need to show that Let Let be an upper triangular element of similar to (such exists because the field of complex numbers is algebraically closed). So

for some invertible element Clearly we can write where is diagonal and is strictly upper triangular. Notice that is nilpotent because all of its eigenvalues are zero (because its eigenvalues are the entries on the main diagonal and those entries are all zero). So

and Thus, in order to complete the solution, we only need to show that Clearly because Let be the -entry of Since we have and thus

So, in order to show that we only need to show that for all and to do that, just write

and see that

So both and are in and therefore by