1.i.10 Linear spaces
condition: Prove that vectors of the form \( \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \) form a linear subspace in the space \( \mathbb{R}^{4} \). Find the basis and dimension of this subspace. Complement the basis of a subspace to the basis of the entire space. Find the transition matrix from the canonical basis of the space \( \mathbb{R}^{4} \) to the constructed basis. \[ \left(x_{1}, x_{2}, x_{3}, x_{4}\right)=(a-b+3 c,-2 b, c, a+2 b) \text {. } \]
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.