Projections and adjoints

Let V be a vector space and T:VV be a linear transformation.

  1. Prove that if T is a projection (i.e., T2=T), then V can be decomposed into the internal direct sum V=null(T)range(T).
  2. Suppose V is an inner product space and T is the adjoint of T with respect to the inner product. Show that null(T) is the orthogonal complement of range(T).
  3. Suppose V is an inner product space and T is an orthogonal projection, i.e., a projection for which the null space and range are orthogonal. Show that T is self adjoint.