Let \(m\) and \(n\) be positive integers and \(F\) a field. Let \(f_1, \ldots, f_m\) be linear functionals on \(F^n\). For \(\alpha\) in \(F^n\) define \[ T\alpha = (f_1(\alpha), \ldots, f_m(\alpha)). \] Show that \(T\) is a linear transformation from \(F^n\) into \(F^m\). Then show that every linear transformation from \(F^n\) into \(F^m\) is of the above form, for some \(f_1, \ldots, f_m\).
Solution
For \(\alpha,\beta\in F^n\) and \(c\in F\), \[ T(\alpha+c\beta) =(f_1(\alpha+c\beta),\ldots,f_m(\alpha+c\beta)) =T\alpha+cT\beta, \] so \(T\) is linear.
Conversely, let \(T:F^n\to F^m\) be linear. For \(1\le i\le m\), let \(\pi_i:F^m\to F\) be the \(i\)th coordinate functional and put \[ f_i=\pi_i\circ T. \] Each \(f_i\) is a linear functional on \(F^n\), and by construction \[ T\alpha=(f_1(\alpha),\ldots,f_m(\alpha)). \] Thus every linear transformation \(F^n\to F^m\) has the stated form.