A 2 × 2 matrix B is upper triangular if

A typical matrix from UT₂ looks like

where a, b, c ∈ C are arbitrary scalars. Observing this we can then write

which says that

is a spanning set for UT₂ (Definition SSVS). Is R is linearly independent? If so, it is a basis for UT₂. So consider a relation of linear dependence on R,

From this equation, one rapidly arrives at the conclusion that α₁ = α₂ = α₃ = 0. So R is a linearly independent set (Definition LI), and hence is a basis (Definition B) for UT₂. Now, we simply count up the size of the set R to see that the dimension of UT₂ is dim (UT₂) = 3.