Problem 1 Let v₁ = (2, −3, 1), v2 = (1, 1, 1), v3 = (4, 1, −5) be a basis for R³. Write the following vectors as a linear combination of the basis vectors av₁ + bv2 + cv3 1. u =(1,2,-1) 2. u =(27,2,7) 3. u =(1,0,0) Problem 2 Let P be the vector space of all polynomials. The linear map T : P → P is defined as T(f(x)) = xxx f(x). Find examples of its eigenvalues and eigenvectors. Problem 3 Prove that the following matrix M in diagonalizable and D = P-¹MP where D is some diagonal matrix. Find the matrices P, D. 4 1 M = 25 -2 1 1 2