dim of quotient spaces Consider $V=Mat_n(K), n \ge 1$ and  3 subspaces $U_1,U_2,U_3 \subset V$ respectively the scalar, diagonal and upper triangular matrices. Calculate dimension of $U_3/U_1 \ \ U_3/U_2  \ \ V/U_1  \ \ V/U_2  \ \ V/U_3$.
I know that  $\dim(V/U)=\dim(V)-\dim(U)$. Scalar matrix has the same dim as V. What about the rest?

The dimension, in this case, is the number of different not identically zero elements from $K$ that show up in your three types of matrices. For "scalar" matrices, there is a single number repeated along the diagonal (and you don't count the surrounding $0$s), so $\dim U_1 =1$. For diagonal matrices, there are as many different not identically zero elements as entries on the diagonal, and this is $n=\dim U_2$. Finally, for triangular matrices there are $n + (n-1) + (n-2) + \cdots +1$ different not identically zero elements, which gives $\dim U_3 = \frac {n(n+1)} 2$. Of course, $dim V=n^2$ and you do the rest.
