Answer Happy

1. In each of the following cases a set of objects is given, together with operations of addition and scalar multiplication. Which of these are vector spaces? Justify your answers. (a) The set of all ordered pairs of real numbers (x,y) with the operations: (x,y) + (a,b) - (x +a, y + b) and (x,y) - (24r, 2Xy). (b) The set of all ordered pairs of real numbers of the form (,y), where 120with the usual operations of addition and scalar multiplication. (e) The set of all functions : R + R such that (2)=0 with the usual operations (as defined in Example 2.1.3). (d) The set of all positive real numbers with the operations 1+ y = ry and Ar= (e) All polynomials of degree 3, with the usual operations (as defined in Example 2.1.8).