A basis of a polynomial space

Express in the canonical basis.
$$1 = 1, \qquad X-1 = X-1, \qquad (X-1)^2 = X^2-2X+1.$$
The matrix of the new family in the basis $(1,X,X^2)$ is
$$P = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{pmatrix}.$$

Determinant. $P$ is upper triangular with diagonal entries $1,1,1$, so $\det P=1\neq0$. The family is a basis.

Dimension. $\dim\mathbb{R}_2[X]=3$ and the family has $3$ independent vectors, so it is a basis.

Conclusion. $(1, X-1, (X-1)^2)$ is a basis of $\mathbb{R}_2[X]$.