1.ya.11 Linear spaces
Condition: Let \( M- \) be a set of polynomials \( p \in \mathbb{P}_{n} \) with real coefficients that satisfy the specified conditions. Prove that \( M \) is a linear subspace in \( \mathbb{P}_{n} \), find its basis and dimension. Complete the basis \( M \) to the basis of the entire space \( \mathbb{P}_{n} \). Find the transition matrix from the canonical basis of the space \( \mathbb{P}_{n} \) to the constructed basis. \[ n=4, \quad p(t) \in M, \quad p(-2)=p(3)=0 \text {. } \]
Linear spaces, subspaces. Investigation of given sets with operations defined on them to compose a linear space. Axioms of linear spaces. Linear spaces of polynomials, matrices, vectors, functions and numbers.