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

4b-c+ 3d -6a -b + 2d with respect to standard ordered bases of P, and M2x2 (R). Also compute L (5x'-*' + 3x + 2) using this matrix. 6. (a) Let T:R'→R2 be the liner transformation given by Find the matrix of T with respect to the ordered bases B = {[1,-3,2], [-4,13,-3], [2,-3,20] for R° and C=(-2,-1] [5,3]} for R'. (b) Consider the linear transformation Find a basis for ker(T) and range(T). Also, verify the Dimension theorem. Consider the linear transformation T: R' → R? Show that L is one-to-one but not onto. 1. (a) If a and b are unit vectors in R". Then prove that -1≤ ab≤1. (b) Solve the following system of linear equations using Gaussian-Elimination method. Examine the consistency of the system and if consistent, find the complete solution set. (c) Solve the following system of linear equations using Gauss-Jordan method: 2. (a) Define rank of a matrix. Find the rank of following matrix by reducing into its reduced row echelon form: A = (b) Find the characteristic polynomial of the matrix: Also, find all the eigenvalues and eigenspaces for A. (c) Let A = Show that matrix A is not 3. (a) Let V be a vector space over R. Show that: (ii) If av = 0v then either a = 0 or V= Ov where aeR,veV and Ov denotes the zero vector of V. (b) Define a subspace. Show that the set S= {peps {peps = 0) is a subspace of p, under the usual operations. Examine whether the set S= ([1,2,l), (2,3,1), [-1,2,-3]} forms a basis for R' or not. 4. (a) Define linear independence and linear dependence for a finite set of vectors. Determine whether the set S = {[1,9,-2), [3,4,5), I-2,5,-7]) is linearly independent or not. (b) Find a basis and dimension of the subspace W of R', where W={XeR |AX=0}and (c) Use the simplified span method to check whether the set {[1,3,-1], (2,7,-3), (4,8,-7]) spans R' or not 5. (a) Let T: V → W be a linear transformation from a vector space V to a vector space W. Define kernel of T and prove that kernel of T is a subspace of V. (b) Determine which of the following functions are linear transformations: (ii) f: Mmxn (R) → Maxm (R) given by f(A) = A for all A eMmxn (R). (c) Find the matrix for the linear transformation