Let U, V, W be vector spaces over the same field, and let f, g be linear maps as follows: Uiviw, with the property that g n f = . 1. Prove that Im(f) E Null(g}, and give an explicit example of vector spaces and linear maps as above, with the property that the Im(f) is a proper subspace of Numg} and Null(g} is a proper subspace of V. 2. Suppose that Q : V > V is a linear operator on a finite dimensional vector space, and suppose Q2 = [1. Suppose further that Nul](Q} = 1111(9). Prove that dint V must be even. Provide an example of such a linear map