Question

Prove or disprove the following. (a) R is a field. (b) There is an additive identity for vectors in R^n. (If true, what is it?)........

1. Prove or disprove the following. (a) R is a field (b) There is an it?) additive identity for vectors in R". (If true, what is (c) There is a is it? multiplicative identity for vectors in R". (If true, what (d) For , , (e) For a, bE R and E R", a(b) = (ab)

Answer 1

(1)(a)R is a field The set of rati onal numbers satisfies ther following properties: [Commutativity of addition] [Associativity of addition] (i)x + y y +x (ii)(x+ y) r (y+z) +Z = (iii) There is an zero element i.e. 0 E R satisfying a0 0 0+a for all a e R Existence of additive identity (iv)There is an element i.e. - xeR for all xe R (-x) 0 for all xeR [Existence of additi ve inverse] (v)xxy yxx (vi) (xx y)xz xx (yx z) (vii)( Commutativity of multi pli cation Associativity of multiplicati on] [Distributivity X X yx z xz yz (viii) There is an element 1e R satisfing Existence of multiplicative identity 1x a a for all a e R (ix) There is an element a e R satisfing axa 1 for all aeR- 0} Existence of multiplicative inverse R is a field.

(b)There is an additive identity for vectors in R" 0 eR" The additive identity is 0 0 a2 Hence Proved. (c)There is a multiplicative identity for vectors in R" The additive identity is 1 a2 x1 Hence Proved. (d)For ,v,u = + & Since vector addition is commutative the above result is true.

(e) For a,be R and v e R",a(bv)= (ab)v a(b)ab 1 bv bv2 bv abvi abv : abv /2 ab 1 -(ab) Hence Proved Il