Advanced Linear Algebra — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

1. (a) Let Tbe a linear operator on R2 defined by T(a, b) = (a - 3b, 2a + b), and let B = {(1,0), (0,1)} and B' = { (1,1), (1,2)} be the ordered bases of R2. Find the change of coordinate matrix O that change B'-coordinates into B-coordinates. Also, verify that (b) Let V = P,(R), the vector space of all real polynomials of degree less than or equal to 1. For P(x) € V, define f,, f, e V* by f (p(x)) = p(t)at and f2 (p(x)) = 5p (t)dt. Prove that {f,, fz} is a basis for V*, then find a basis for V for which it is the dual basis. (c) Let V denotes a finite-dimensional vector space. Define the annihilator S° of a subset S of V and prove that S° is a subspace of V*. Further, if W is a subspace of V and x € W, then prove that there exists fe W° such that f(x) + 0. 2. (a) LetA = (2: 4) = Ma(C). Determine all the eigenvalues of A. For each eigenvalue i of A, find the set of eigenvectors corresponding to 2. Also, find a basis for C2 consisting of eigenvectors of A. (2.5+5) (b) For A = 2 1)E Mza (B), find an expression for A", where n is a positive integer. (7.5) (c) Let P(R) denote the vector space of all real polynomials of degree less than or equal to 2. Let T be a linear operator on P(R) defined as Prove that P(R) is the direct sum of the eigenspaces of T. (7.5) 3. (a) Let T be a linear operator on a vector space V and z be a nonzero vector of V. Define a T-cyclic subspace of V generated by z. Show that it is a T-invariant subspace of V. Further, for V = P(R), the vector space of all real polynomials of degree less than or equal to 3, T(f(x)) = f" (x), and z = x3, find an ordered basis for the T-cyclic subspace of (b) State the Cayley-Hamilton Theorem for a linear operator on a finite-dimensional vector space V. Let T be the linear operator on P_(R) defined by T(f(x)) = f(0) + f(1)(x +x). Verify the Cayley- Hamilton Theorem for the linear operator T. Find a basis for each generalized eigenspace of La consisting of a union of disjoint cycles of generalized eigenvectors: Then find a Jordan canonical form J of A. (5.5+2) 4. (a) Define the minimal polynomial of a linear operator on a finite-dimensional vector space. Let T be the linear operator on P(R) defined by T(f(x) = -xf" (x) + f(x) + 2f(x). Find the minimal polynomial. (b) Let V = C([0,1]), die vector space of real-valued continuous functions on [0,1]. For f, g e V, define : Prove that V with <.,.> defined above is an inner product space. For f(t) = t and g(t) = e', compute (c) Let x = (2,1 + i, i) and y = (2-i, 2, 1 + 2i) be vectors in C3. Compute <x,y>, ||x|l, Ilyll, and x + yll. Then verify both the Cauchy-Schwarz inequality and the triangle inequality for the given vectors x and y. (4+2+1.5) 5. (a) Let P(R) be the space of all real polynomials of degree less than or equal to 3 with inner product Also, let S = {1, x, x?, x3). Apply the Gram-Schmidt process to the set S to obtain an orthonormal basis for P3(R). (7.5) (b) Find the best-fit linear function for the data