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

1. (a) If x and y are vectors in R", then prove that Also verify it for the vectors x = [2, - 1,3,2] and y = [4,3,2,1] in R4. (b) Solve the following system of linear equations using Gaussian Elimination method: Also give the complete solution set if the system is consistent. (6+1.5) (c) Using the Gauss-Jordan method, find the minimal integer values for the variables that will balance the following chemical equation: 2. (a) Find' the reduced row echelon form B of the following matrix A, keeping : track of the row operations used : (b) Determine whether the vector x = [5,17, - 20] is in the row space of the matrix If so, then express [5,17, -20] as a linear combination of the rows of (5+2.5) (c) (i) Define Eigenvalue and Eigenspace for a square matrix A. (i) Find all the eigenvalues and eigenspaces for A. 3. (a) Define a 'subspace of a vector space over R. Which of the following are subspaces of M, (IR)? Give reasons to support your answer. . (b) Define sum of two subspaces W, and W., of a vector space V(R). Show that W, + W, is the smallest subspace of V containing both W, and W,. (2+5.5) (c) Define Linear SLpan of a set S in a vector space V. Is the linear span of S a subspace of v? Determine if the vector (2, -1,1) is, in the linear span of 4. (a) Define a linearly independent subset`Qfavector space. For k = 0,1,2 ,...., n, pk(x) = xk + xk+1 + . . . + xn. Show that {po(x), p](x) ,..., p](x)} is linearly independent sub+set of the spape of all polynomials. (2+5.5) (b) Let W = {(a], a2, a3, a4, a5): a2 = a4,a] + a3 + a5 = 0} be a subset of R5. Is W a subspace of R5? Find a basis for W. What is dimR W? (2+4+1.5) (c) Let S = {(1,3, -4,2), (2,2, -4,0)3 (1, -3,2, -4), (-1,0,1,0)} be al.subset of R4. Show that S is linearly_ dependent. Write`Aa vector in S as a linear combination of other vectors. (5.5+2) 5. (a) Define a linea.r transformation. Is there any linear transformation T: R3 + R2 such that T(1,0,3) = (1,1) and T(-2,0,-6) = (2,1)? (2+5.5) (b) Let T: nR2 i R3 be defined by T(a, b) = (a-b, a, 2a + b) Let P be the standard ordered basis for R2 and y = {(1,1,0), (0,1,1), (c) (i) Let V and W be vector spaces` and let T : V i W be a linear transformation. Then show that T is one-to-one if and only if N(T) = (ii) Let T: P2(R) i P3(R) be a linear transformation defined by TIT cxtyD -- 2f'tx3 + Show that T is one-one. (a) Let V and W be vector spaces and let T: V i W be a linear. Define inverse of T. If T is invertible linear transformation then show that T-I: W + V is linear. (b) Define isomorphic vector spaces. Which of the following spaces are isomorphic? Justify your answer. (i) Maxz(R) and P, (IR) (c) For the graphic in given figure, use ordinary coordinates to find the new vertices after performing each indicated operations: (i) Rotation about the origin through 0 = 90° (i) Rection about the line y = x. (iii) Translation along the vector [-3, 5] (2.5+2.5+2.5)