Question

Verify this axiom of a vector space.

Vector space:

A subspace of R²: the set of all dimension-2 vectors [x; y] whose entries x and y are odd integers.

Axiom 1:

The sum u + v is in V.

Answer 1

we take two vectors u,v in V then we found their sum.let u=[x;y]

in V so x,and y are odd integers.similaryly let v=[x',y'] so x' and y' are odd integers.so u+v=[x+x',y+y']

sum of two odd integers is even always(let x=2k+1,x'=2m+1 so x+x'=2(k+m)+2=2(k+m+1) hence x+x' is even)

so x+x' is not odd integer hence u+v is not in V.so axiom 1 fails in this case.hence it is not a vectorspace.