Let V be a finite-dimensional vector space over F, and suppose that T ∈ L(V ) is a linear operator having the following property: Given any two bases b and c for V , the matrix M(T, b) for T with respect to b is the same as the matric M(T, c) for T with respect to c. Prove that there exists a scalar α ∈ F such that T = αidV , where idV denotes the identity map on V .

I don't really know how to approach this, and some guidance would be helpful! Thanks.