Let \(T\) and \(U\) be the linear operators on \(\RR^2\) defined by \[ T(x_1, x_2) = (x_2, x_1) \quad \text{and} \quad U(x_1, x_2) = (x_1, 0). \]
How would you describe \(T\) and \(U\) geometrically?
Give rules like the ones defining \(T\) and \(U\) for each of the transformations \((U + T)\), \(UT\), \(TU\), \(T^2\), \(U^2\).
Solution
The map \(T\) interchanges the two coordinates, so geometrically it is reflection across the line \(x_1=x_2\). The map \(U\) is projection onto the \(x_1\)-axis.
For \((x_1,x_2)\in\RR^2\), \[ (U+T)(x_1,x_2)=(x_1+x_2,x_1), \] \[ (UT)(x_1,x_2)=U(x_2,x_1)=(x_2,0), \qquad (TU)(x_1,x_2)=T(x_1,0)=(0,x_1). \] Also \[ T^2(x_1,x_2)=(x_1,x_2), \qquad U^2(x_1,x_2)=(x_1,0). \] Hence \(T^2=I\) and \(U^2=U\).