## linear algebra - direct sums

Let S denote the subset of V= Q^n*n which consists of upper triangular matrices with 1's along the diagonal. For example, if n=3 these are matrices of the form

1 a b
0 1 c
0 0 1

Let B denote the smallest subspace of V containing S.

(a) What is B? (write a description of the matrices in B)
(b) Prove that the set you defined above is a subspace of V.
(c) Fing a subspace A of V so that A+B=V (direct sum). Prove that A is a subspace and that A+B=V (direct sum). (A is called the complement of B)