1.9.8 Linear spaces

Problem: Let \( M \) be a set of polynomials \( P \in \mathrm{P}_{n} \) with real coefficients satisfying the specified conditions. Prove that \( M \) - linear subspace in \( \mathrm{P}_{n} \), find its basis and dimension. Complement basis \( M \) to the basis of the whole space \( P_{n} \). \[ n=3, \quad M=\left\{P \in \mathrm{P}_{3} \mid P^{\prime \prime}(1)+P^{\prime}(0)=0\right\} . \]