Question

8. Let W be the set of all vectors in R3 of the form a(8, 9, 1), where a is a real number. A. Let b and c be arbitrary real numbers such that b(8, 9, 1) and c(8, 9, 1) are in W. Is b(8, 9, 1) + c(8, 9, 1) in W?

Answer 1

Here W = {a(8, 9, 1): a elementof 1R} Because b(8, 9, 1) + c(8, 9, 1) = (b + c) (8, 9, 1) elementof W. [Because b elementof 1R, c elementof 1R Rightarrow b + c elementof 1R] YES. k(b(8, 9, 1)) = k b(8, 9, 1) elementof W [Because k elementof 1R (scalar) & b elementof 1R rightarrow k b elementof 1R] Let V be vector space over a field 1F.A subset W Subsetorequalto V is said to form subspace of V over same field 1F and same operations if (i) forall u_1, u_2 elementof W Rightarrow u_1 + u_2 elementof W (ii) Forall k elementof 1F, v elementof W Rightarrow k v elementof w These two condition is satisfied by (A) & (B). Hence W is subspace of 1R^3.