Chapter 4 - Vector Spaces - 4.2 Vector Spaces - 4.2 Exercises - Page 160: 32

it is a vector space.

Any $3\times 3$ upper triangular matrix takes the form $$\left[ \begin {array}{cccc} a&b&c\\ 0&d&e \\ 0&0&f \end {array} \right], \quad a,b,c,d,e,f\in R$$ It is easy to note that the set of all $3\times 3$ upper triangular matrices contains the zero vector and closed under the standard addition and scalar multiplication. Strightforward calculations show that it is a vector space.