Let \(V\) be the set of pairs \((x, y)\) of real numbers and let \(F\) be the field of real numbers. Define \[ (x, y) + (x_1, y_1) = (x + x_1,\; 0), \qquad c(x, y) = (cx,\; 0). \] Is \(V\), with these operations, a vector space?
Solution
No. There is no additive identity for all of \(V\). If \((e_1,e_2)\) were an additive identity, then for every \((x,y)\) we would need \[ (x,y)+(e_1,e_2)=(x+e_1,0)=(x,y). \] The second coordinate would force \(0=y\) for every real \(y\), which is impossible. Therefore the additive-group axioms already fail, so \(V\) is not a vector space with these operations.