Linear Algbra

Question 2 (a) Consider the vector space R5 equipped with the weighted Euclidean in- ner product with weights wl = 2,1112 = 1,103 = 3,1124 = Law, = 1. Find a basis for the orthogonal complement (with respect to this inner product) of the subspace span(v1,v2,v3) where v1 = (1,2,3,1,0), v2 = (0,-3,1,-4,6), v3 = (2,7,5,2,6). Show your working. [6 Marks] (b) Prove or disprove: if {u, v} is a basis in R2 then there exist positive weights 101,102 6 R such that u and v are orthogonal with respect to the weighted Euclidean inner product with weights whwg. [4 Marks] (c) Consider the vector space C'[0, 7r] of all continuous functions on the interval [0, 7r], equipped with the inner product = /\" fg are where f = f(m),g 2 9(32). Find an orthonormal basis of the subspace of C[0,1r] spanned by the vectors v1 = 1, V2 = m, V3 2 sin(a:). Show your working. [7 Marks] (d) Do there exist symmetric 3 x 3 matrices A and B such that i. A has eigenvalues A1 = 3. A2 = 0, and A3 = 7 and correSponding eigenvectors v1 = (0,1,1). v2 = (1,0,0). v3 = (1,1,1)? ii. B has eigenvalues A1 = 2. A2 = 5, and A3 = 6 and corresponding eigenvectors v1=(O,1,1),v2 = (1,0,0), V3 2 (0,1,1)? In each case, either find such a matrix or explain why it does not exist. [8 Marks]