Let \(V\) be the set of all pairs \((x, y)\) of real numbers, and let \(F\) be the field of real numbers. Define \[ (x, y) + (x_1, y_1) = (x + x_1,\; y + y_1), \qquad c(x, y) = (cx,\; y). \] Is \(V\), with these operations, a vector space over the field of real numbers?
Solution
No. The scalar-multiplication axiom \[ 1\cdot v=v \] fails in general. Indeed, for \(v=(x,y)\) the defined operation gives \[ 1\cdot(x,y)=(x,y), \] so this axiom does hold; however distributivity over scalar addition fails. Take \(v=(0,1)\) and scalars \(1,1\). Then \[ (1+1)v=2v=(0,1), \] whereas \[ 1v+1v=(0,1)+(0,1)=(0,2). \] Thus \((a+b)v=av+bv\) is false, so these operations do not make \(V\) a real vector space.