let T denote a linear operator on a vector space V. suppose that every subspace of V is invariant under T. prove that T is a scalar multiple of the identity map.
please help me.
Step 1: Let x be a nonzero vector in V. Then the one-dimensional subspace spanned by x is invariant under T. So , for some scalar that (possibly) depends on x.
Step 2: Let x and y be linearly independent vectors in V. Then . Deduce from this that . Thus the constant c is in fact the same for every vector, and hence T is c times the identity.