1. The problem statement, all variables and given/known data Suppose T is contained in the set of linear transformations from V to V. Prove that the intersection of any collection of subspaces of V invariant under T is invariant under T. 2. Relevant equations 3. The attempt at a solution Choose a basis for V. This basis is <v1.....vn>. Two possible subspaces for V are A and B with basis <v1....vi> for A and <vi-1.....vn> for B (2[tex]\leq[/tex]i[tex]\leq[/tex]n) and we will assume that both are invariant under T. A basis for the intersection of the bases is <vi-1, vi>. Because the subspaces A and B are invariant under T, and vi-1+vi is a linear combination within the span of both invariant subspaces under T, vi-1+vi is mapped from A to A and from B to B thus it goes from A and B to A and B, the intersection of both subspaces.