Consider the polynomials f(t) = t + 1 and g(t) = (t + 2)(t + k), where k is an arbitrary constant. For which values of the constant k are the three polynomials f(t), tf(t) and g(t) a basis of P2?
This will happen when the matrix of coefficients
$\displaystyle \left(
\begin{array}{lll}
1 & 0 & 2 k \\
1 & 1 & k+2 \\
0 & 1 & 1
\end{array}
\right)$
is nonsingular. Evaluate its determinant, and find the values of $\displaystyle k$ for which it is nonzero.