MAT2611 Assignment 2 2024 - DUE 3 May 2024

MAT2611 Assignment 2 Due date: Friday, 03 May 2024 ... 100 % TRUSTED workings, explanations and solutions. For assistance call or W.h.a.t.s.a.p.p us on +/ 2/ 5/ 4 /7 /7 /9 /5 /4 /0 /1 /3 /2 . ASSIGNMENT 02 Due date: Friday, 03 May 2024 Problem 5. Determine whether each set equipped with the given operation is a vector space.For those that are not vector space identify the vector space axioms that fail. (1) The set U = f(x; 0) 2 R 2g with the standard operations on R 2: (2) The set V = f(x; y) 2 R 2 : y 0g with the standard operations on R 2: (3) The set W = f(x; y) 2 R 2 : x + y = 0g with the standard operations on R 2: (4) The set X = f(x; y) 2 R 2g with the standard vector addition but with scalar multiplication de…ned by k (x; y) = k 2x; k2y : (5) The set of all 2 2 matrices Y = a b c 0 : a; b; c 2 R with the standard matrix addition and scalar multiplication. [10 marks] Problem 6. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on u = (u 1 ; u2 ; u3) and v = (v 1 ; v2 ; v3) : u + v = (u 1 + v 1 ; u2 + v 2 ; u3 + v 3) ; ku = (ku 1 ; ku2; 0) : (1) Compute u + v and ku for u = ( 1; 2; 3) ; v = (2; 3; 1) and k = 2: (2) Determine whether the Axioms 7, 8, 9 and 10 hold. [10 marks] Problem 7. Let V be a vector space, u a vector in V; and k a scalar. Then show that if ku = 0; then k = 0 or u = 0: [10 marks] Problem 8. Let 1 and 1 denote two distinct objects, neither of which is in R. De…ne an addition and scalar multiplication on R[ f1g [ f1g. Speci…cally, the sum and product of two real numbers is as usual, and for k 2 R de…ne k1 = 8 : 1 if k 0 0 if k = 0 1 if k 0; k ( 1) = 8 : 1 if k 0 0 if k = 0 1 if k 0; k + 1 = 1 + k = 1; k + ( 1) = 1 + k = 1; 1 + 1 = 1; ( 1) + ( 1) = 1; 1 + ( 1) = 0: [10 marks] Show that R [ f1g [ f1g is not a vector space over R. [Total: 40 marks] – End of assignment-