Advanced Linear Algebra — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Name of, the Course : B.Sc. (H) Mathematics (NEP-UGCF 2022) (a) Let B and f' be two ordered bases for a finite-dimensional vector space V. (i) Define change of coordinate matrix/Q that changes f'-coordinates into B-coordinates. (i) Show that for any 0 EV, = QUIB. (iii) Let B = {(1,1), (1, - 1)} and ß' = {(2,4), (3,1)} be two ordered bases of R'. Taking = (2,3), verify that [0] =Q[] (b) Let T: P, (R) → R? be a linear transformation given by Let B and y be standard ordered bases for P(R) and R? respectively. Verify that [TIg* = (171) . (c) Define annihilator V7° of 'a subset W of a vector space V. Show that Wº is a subspace of V. Prove that if W is a subspace of V, then dim(W) + dim(W°) = dim (V). 2. (a) Define eigenvalues, eigenvectors and characteristic polynomial of a linear operator T defined on an n-dimensional vector space V.. Let T be the operator on R? that rotates the plane through an angle 0. (0 < 0<1). Obtain the characteristic polynomial of T and hence deduce that T has no eigenvalues and eigenvectors. (b) Show that the characteristic polynomial of any diagonalizable linear operator splits. Is the converse true? Justify. (c) Let W1. W2,..., Wk be subspaces of a finite-dimensional vector space V. Show that V = W1 $ W2 # ... * Wr if and only if each vector v € V. can be uniquely written as 0 = 21 + 02+... + Vk for some v; E Wi. For which of the following linear operator T' on the vector space V., determine whether the given subspace W is T-invariant subspace of V: (b) Let T be a linear operator on a finite dimensional vector space V and let W be a T-invariant subspace of V. Prove that if the characteristic polynomial of T splits, then the characteristic polynomial of Tw also splits. (c) Compute the minimal polynomial of the matrix A, (a) Let x = (2,1 + i, - i) and y = (2 + i,2,1 - 2i) be vectors in C3. Compute (,), ||, |||| and || t y. Then verity the Cauchy-Schwarz inequality and the triangle inequality for above vectors x and y. (b) Prove that if V is an inner product space, then (x; y) = Hellill if and only if one of the vectors x or y is a multiplé of the other. (c) Ușing Gram-Schmidt process, obtain an orthonormal basis ß for span(S) where S. = {(1,0,1), (0,1,1), (1,1,0)). Also, compute the Fourier coefficients of x = (1,1,1) with respect to B. (a) Write the definition of the orthogonal complement of a set S in an inner product space V.. Let W = span[(i, 0,1)} in C3. Find orthonormal bases for (b) For each of the following inner product spaces V and linear operators T on V, evaluate T* at the given vector in V: (c) Find the minimal solution for the following system of linear equations x+Zy+z= (a) Let T be a linear operator on a finite dimensional complex inner product spacé V. Prove that T is normal if and only if there exists an orthonormal basis for V consisting of eigenvectors of T. (b) For the given matrix A, find an orthogonal matrix P and a diagonal matrix D such that P*AP = D, where