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

Non-Major / Minor b. Show that the set of vectors of the form [2a - 3b, a - 5c, a, 4c - b, c] in RS forms a subspace of R' under usual operations. c. Prove or disprove : i) The set of polynomials of degree 5 is a vector space under the operations of usual addition and scalar multiplication of polynomials. ii) The set of singular 2*2 matrices is a vector space under the operations of usual addition and scalar multiplication of matrices. 4. a. Use simplified span method to find a simplified general form for all vectors in span(S), where b. Use independence test method to determine whether the set is linearly independent or a linearly dependent subset of R3? c. Determine whether the subset of P2 forms a basis of P2, the set of all polynomials of degree ≤ 2. 5. a. State True or False for the following statements. Give justification for your choice: (i) A linear transformation T from vector space V to W, maps linearly independent subset S of V to a linearly independent subset of W. (3.5) (ii) The function L: R3 → R3 given by L(x, y, z) = (x + 1, y - 2, z + 3) is a linear transformation. b. Prove that the inverse of a linear transformation L, (if it exists), is also a linear transformation. c. Find the matrix for the linear transformation T: P3 (IR) → R defined by with respect to the standard ordered bases for vector spaces P3 (R) and R' (R). 6. a. Find the kernel and range of the transformation T: P3 (IR) → R3 defined by Also, verify the Dimension Theorem for T. b. Let T: R2 → R2 be a linear transformation that performs a counterclockwise rotation through an angle of 30°. Find the matrix representation of T with respect to ordered basis B = {[1,1], [2,3]) for R2. c. Let T: P3 (R) → P2 (R) be a linear transformation given by Check whether T' is an invertible linear transformation or not?