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Practice Exam #2 Math 2270, Spring 2005 Problem 1. Let P2 be the space of all polynomials of degree ≤ 2 and consider the following two bases: U = (1, t, t2 ) and B = (1, t − 1, (t − 1)2 ). If T : P2 → P2 is the linear transformation given by T (f (t)) = f (2t − 1), find the following: (a) The matrix A of T with respect to the basis U. (b) The matrix B of T with respect to the basis B. (c) The change of basis matrix S from the basis B to the basis U. (d) Verify the formula SA = BS. (e) Find the change of basis matrix from U to B. (f) Is T an isomorphism? Problem 2. Let V be the subspace of R3 given by V = span(v~1 , v~2 ) where 4 25 0 v~1 = 0 and v~2 = 3 −25 (a) Find an orthonormal basis U of V . 4 25 0 . (b) Write a QR-factorization of the matrix A = 0 3 −25 (c) Find the matrix of the orthogonal projection on V with respect to the basis U. (d) Find the matrix of the orthogonal projection on V with respect to the basis B = (v~1 , v~2 ) in terms of the matrix A. Do not simplify! 49 25 ~ ~ (e) Find the least square solution of the system A~x = b, where b = 0 . 7 25 Problem 3. Prove that if W ⊆ V are two linear spaces, then dim(W ) ≤dim(V ). Problem 4. Give the definition of the following: (a) Orthogonal complement. Give an example. (b) Orthogonal matrix. Give an example. Problem 5. True or false? Justify all your answers. (a) If A is an invertible matrix, then the equation (AT )−1 = (A−1 )T must hold. (b) Every non-zero space has an orthonormal basis. (c) Im(A)=Im(AT A) holds for all matrices A, n × m. (d) The kernel of the linear transformation T (f ) = f ′ from P to P is {0}. (P is the set of all polynomials.)