1.9.1 Linear spaces

Problem: Let \( M \) - be a set of polynomials \( p \in P_{n} \) with real coefficients satisfying the specified conditions. Prove that \( M \) - is a linear subspace in \( P_{n} \), find its basis and dimension. Complement the basis of \( M \) to the basis of the entire space \( P_{n} \). Find the transition matrix from the canonical basis of the space \( P_{n} \) to the constructed basis. \[ n=3, M=\left\{p \in P_{3} \mid p^{\prime \prime}(a)+p^{\prime}(0)=0\right\} \text {. } \]