If \(F\) is a field, verify that \(F^n\) is a vector space over \(F\), where \(F^n\) is the set of \(n\)-tuples of elements of \(F\) with componentwise addition and scalar multiplication.
Solution
Let \(x=(x_1,\ldots,x_n)\), \(y=(y_1,\ldots,y_n)\), \(z=(z_1,\ldots,z_n)\) in \(F^n\) and \(a,b\in F\). Every vector-space axiom follows coordinatewise from the corresponding field axiom in \(F\). For example, \[ x+y=(x_1+y_1,\ldots,x_n+y_n), \] so addition is associative and commutative because addition in \(F\) is; the zero vector is \((0,\ldots,0)\) and \(-x=(-x_1,\ldots,-x_n)\). Likewise \[ a(x+y)=ax+ay, \qquad (a+b)x=ax+bx, \qquad (ab)x=a(bx), \qquad 1x=x \] hold coordinatewise. Hence \(F^n\) is a vector space over \(F\).