Determine whether the given set together with the specified

Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space over the indicated Z_p. If it is not, select all of the axioms that fall to hold. (Let u, v, and w be vectors in the vector space V, and let c and d be scalars.) The set of all vectors in Z_2^n with en odd number of 1s, over Z_2 with the usual vector addition and scalar multiplication All of the axioms hold, so the given set is a vector space. u + v is in V u + v = v + u (u + v) + w = u + (v + w) There exists an element 0 in v, called a zero vector, such that u + 0 = u For each u in V, there is an element -u in V such that u + (-u) = 0 cu is in V c(u + v) = cu + cv c(du) = (cd) u 1u = u

Solution

We know that : Z ₂n == { 0,1,2, -------( 2ⁿ -1) } the vectors u ,v ------are in this set

Z ₂= {0,1} the scalars c,d can take the values 0 or 1

Let u = 2ⁿ-1 , v =2ⁿ-1

then u+v = 2(2ⁿ-1) = 2ⁿ⁺¹ -2 is not an element in the set

ie WRT the opertion + the closure property is not satisfied .

All the other properties satisfy except property (1)

hence the set is not avector space