Linear Algebra Done Right · Problem 5C.14

Eigenvalues, Eigenvectors, and Invariant Subspaces — Sheldon Axler

Mathematics — Linear Algebra · upper-triangular-matrices

Problem

Suppose

V

is finite-dimensional and

T \in \mathcal{L}(V)

. Prove that

T

has an upper-triangular matrix with respect to some basis of

V

if and only if the dual operator

T'

has an upper-triangular matrix with respect to some basis of the dual space

V'