in the the standard basis {1,t,t^{2}} (note: the order matters here), the coordinate form of (t-1)^{2}is (1,-2,1), and the coordinate form of (-2t)(1-t) = 2t(t-1) = 2t^{2}-2t is (0,-2,2).

EDIT: according to wikipedia, your second basis vector should be 2t(1-t), which would be (0,2,-2).

Bernstein polynomial - Wikipedia, the free encyclopedia

what i get for the matrix is: