Describe explicitly (as in Exercises 1 and 2) the linear transformation \(T\) from \(F^2\) into \(F^2\) such that \(T\epsilon_1 = (a, b)\), \(T\epsilon_2 = (c, d)\).
Solution
Every \((x,y)\in F^2\) has the basis expansion \[ (x,y)=x\epsilon_1+y\epsilon_2. \] By linearity, \[ T(x,y)=xT\epsilon_1+yT\epsilon_2 =x(a,b)+y(c,d). \] Hence \[ \boxed{T(x,y)=(ax+cy,\;bx+dy)}. \] Equivalently, relative to the standard bases, \[ [T]=\begin{bmatrix}a&c\\b&d\end{bmatrix}. \]