Lecture 1: n-Dimensional Vector Spaces | Calculus Revisited: Multivariable Calculus | Mathematics | MIT OpenCourseWare

Let T be a linear operator on a finite-dimensional vector space V, and suppose there exists an ordered basis βfor V such that [T]β is an upper triangular matrix. (a) Prove that the characteristic polynomial for T splits. (b) State and prove an analogous result for matrices. The converse of (a) is treated in Exercise 32 of Section 5.4. | Numerade

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