Question

show that the set x with the given operations fails to be a vector space by identifying all axioms that hold and fail to hold: The set x={(a,1):a is a element of R} c R^2 with vector addition defined by (a,1)+(b,1)=(a+b,1) and scalar multiplication defined by k*(a,1)=(k^2 a, 1)

Answer 1

Check that, for any scalars \(a,b\,\) and any vector \(\mathbf{u}\), \((a+b)\mathbf{u}=a\mathbf{u}+b\mathbf{u}\). If this does not hold, then the given field is not a vector space.