Introduction to Magnetism, Force on Dielectric in Charged Capacitor Class 12 Physics | JEE Main 2023

SOLVED: Let V be a finite-dimensional vector space over field F (here F can be any field).An operator P : V -> V is called idempotent if P^2 = P. Show that for such P, we have a decomposition V = Im(P) ⊕ Ker(P); direct sum of the image of P and its kernel.(b) Let W ⊆ V be a subspace. An operator Pw : V -> V is called a projector onto W if its image Im(Pw) = W and Pw acts on W as the identity operator Idw, i.e. Pw(w) = w for every w ∈ W. Show that the following conditions are equivalent (na