Prove that R^n,n , the set of all real nxn matrices forms a vector space over R of dimension n^2. Determine which of the following subsets of R^3,3 are subspaces and (when possible) find a basis for the subspace and determine the dimension.
(a)The set of all diagonal matrices A with aij (i and j supposed to be subscripts)not equal to zero for i not equal to j.
(b)The set of all matrices A with detA=0
(c)The set of all upper triangular matrices A with aij=0 (again, i and j being subscripts) for i>j
(d)The set of all matrices with traceA=0