show that for any two upper triangualr matrices and scalar the matrix is again upper triangular. this shouldn't take more than 2 minutes!

let be the standard basis for then is a basis for

(2) Give a basis for

by (2) it's clear that

(3) What is dim

if then every diagonal matrix would be skew symmetric and upper triangular. thus and hence the sum wouldn't even be direct! however, since in this case(4)Let denote the subspace off all skew-symmetric matrices (satisfying ). Show that if and only if is not characteristic 2.

and we'll have and thus (note that the sum is not direct here!)

if then the diagonal of every element of would be all 0. thus in this case so the sum is direct. we also have therefore

and so