So is simply the set of all nxn real upper triangular matrices. Show now that this set is closed
under sum of matrices and multiplication by scalar (piece of cake).
For its dimension: by what elements is any element of the above set uniquely and completely determined? In how many
ways can you choose lin. ind. real numbers for these elements? Well, this is the dimension, and to choose now
2 (or 100000) different basis for it is another piece of cake.