1. The problem statement, all variables and given/known data Let W be a 1-dimensional subspace of V that is A-invariant. Show that every non zero vector in W is a eigenvector of A. [A element of Mn(F)] 3. The attempt at a solution We know W is A-invariant therefore for all w in W A.w is in W. W is one dimensional which implies to me that A must therefore be a one by one matrix with an entry from F. Is this a correct assumption? If so then A.w=λ.w where λ is an element of F which implies that all w in W are eigenvectors of A. I'm new to this sort of linear algebra and therefore can't tell if I've made a blatant mistake?