Let \(E\) be a projection of \(V\) and let \(T\) be a linear operator on \(V\). Prove that the range of \(E\) is invariant under \(T\) if and only if \(ETE = TE\). Prove that both the range and null space of \(E\) are invariant under \(T\) if and only if \(ET = TE\).
Solution
1. Part 1: Show \(\operatorname{im}(E)\) is \(T\)-invariant if and only if \(ETE = TE\):
1.1. Recall that for a projection \(E^2 = E\), the range is characterized by \(\operatorname{im}(E) = \{v \in V \mid Ev = v\} = \{Ev \mid v \in V\}\).
\(E(Ev) = E^2 v = Ev\).
1.2. Forward direction (\(\implies\)): Assume \(T(\operatorname{im}(E)) \subseteq \operatorname{im}(E)\).For any \(v \in V\), \(Ev \in \operatorname{im}(E)\).By \(T\)-invariance, \(T(Ev) \in \operatorname{im}(E)\).Since \(E\) acts as the identity on \(\operatorname{im}(E)\), \(E(TEv) = TEv\).Thus \(ETE v = TE v\) for all \(v \in V\), so \(ETE = TE\).
\(E|_{{\operatorname{im}(E)}} = \operatorname{id}\).
1.3. Reverse direction (\(\impliedby\)): Assume \(ETE = TE\).For any \(w \in \operatorname{im}(E)\), \(Ew = w\).Then \(Tw = T(Ew) = (TE)w = (ETE)w = E(TEw) = E(Tw) \in \operatorname{im}(E)\).Thus \(T(\operatorname{im}(E)) \subseteq \operatorname{im}(E)\).
\(\operatorname{im}(E)\) is the image of \(E\).
2. Part 2: Show \(\operatorname{im}(E)\) and \(\ker(E)\) are both \(T\)-invariant if and only if \(ET = TE\):
2.1. The complementary projection is \(I - E\), which satisfies \((I - E)^2 = I - E\), \(\operatorname{im}(I - E) = \ker(E)\), and \(\ker(I - E) = \operatorname{im}(E)\).
properties of complementary projections.
2.2. Applying Part 1 to \(I - E\), \(\ker(E) = \operatorname{im}(I - E)\) is \(T\)-invariant if and only if:\[ (I - E)T(I - E) = T(I - E). \]
1 applied to the projection \(I - E\).
2.3. Expand both sides of the condition in 2:\[ (I - E)(T - TE) = T - TE - ET + ETE. \]Thus:\[ T - TE - ET + ETE = T - TE \iff ETE = ET. \]Therefore \(\ker(E)\) is \(T\)-invariant if and only if \(ETE = ET\).
algebraic expansion and cancellation of \(T - TE\).
2.4. Combining the two invariance conditions:- \(\operatorname{im}(E)\) is \(T\)-invariant \(\iff ETE = TE\).- \(\ker(E)\) is \(T\)-invariant \(\iff ETE = ET\).Therefore, both \(\operatorname{im}(E)\) and \(\ker(E)\) are \(T\)-invariant if and only if \(TE = ETE = ET \iff ET = TE\).
1 and 3.
3. Conclusion:\(\operatorname{im}(E)\) is \(T\)-invariant \(\iff ETE = TE\), and both \(\operatorname{im}(E)\) and \(\ker(E)\) are \(T\)-invariant \(\iff ET = TE\). Q.E.D.
1 and 2.