Linear Algebra Problem

"On sheet 1, we saw that the set U of upper triangular matrices in Mn (C) is a vector space
over C in its own right. Write down a basis for U , and hence calculate the dimension of U .
(Hint: The standard basis for Mn (C) which we saw in lectures contains a basis for U .)"

How would I go about doing this?

What's the obvious basis for all of Mn ( C)? A basis for U should be obvious then.
The number of basis vectors is then the dimension of U.

I recall dimension of upper triangular matrices is n(n+1)/2. One choice for the first vector of your base : the first coordinate, rests are zero. Then 2 choices for the next vector : the two firsts coordinate, rests are zero. Etc. They are linearly independant. Hence dimension.