Math 110 Final Exam Professor K A Ribet December 16 2003 Please put away all books calculators electronic games cell phones pagers mp3 players PDAs and other electronic devices You may refer to a single 2 sided sheet of notes Please write your name on each sheet of paper that you turn in don t trust staples to keep your papers together Explain your answers in full English sentences as is customary and appropriate Your paper is your ambassador when it is graded 1 Let A be an n n matrix Suppose that there is a non zero row vector y such that yA y Prove that there is a non zero column vector x such that Ax x Here A x and y have entries in a field F This is a restatement of problem 6 on HW 14 What is given is that 1 is an eigenvalue of the transpose of A It follows that 1 is an eigenvalue of A this gives the conclusion 2 Let A and B be n n matrices over a field F Suppose that A2 A and B 2 B Prove that A and B are similar if and only if they have the same rank This is problem 10 in HW 14 If A and B are similar then they certainly have the same rank Indeed we saw early on in the course that the rank of a matrix does not change if you multiply it on either side by an invertible matrix The harder direction is the converse If T 2 T where T is a linear operator on a vector space V then we know well that V is the direct sum of the null space of T and the space of vectors that are fixed by T See e g problem 17 on page 98 of the textbook The dimension of this latter space is clearly the rank of T Choose a basis v1 vr for the range of T and a basis vr 1 vn for the null space of T The matrix of T with respect to the basis v1 vn is the direct sum of the r r identity matrix and the n r n r zero matrix Taking now T LA we see that A is similar to a matrix that depends only on its rank If A and B have the same rank they are each similar to a common matrix so they re similar to each other 3 Suppose that T V V is a linear transformation on a finite dimensional real inner product space Let T be the adjoint of T Show that V is the direct sum of the null space of T and the range of T The rank of T coincides with the rank of T for various reasons For example in matrix terms this equality is the statement that a square matrix and its transpose have the same rank Hence the dimensions of the null space of T and the range of T add up to the dimension of V This necessary condition for V to be the indicated direct sum is a good sign Also it means that V is the direct sum of the two spaces if and only if V is the sum of the two spaces and that V is the direct sum of the two spaces if and only if the spaces have zero intersection in V Let us prove the latter statement Suppose that T v 0 and that v T w for some w We need to prove that v 0 It is enough to show that hv vi 0 But hv vi hv T w i hT v wi h0 wi 0 4 Let A be a symmetric real matrix whose square has trace 0 Show that A 0 Use the fact that A is similar to a diagonal matrix If B is similar to A then B has the same trace as A also B 0 if and only if A 0 Hence we can and do assume that A is a diagonal X matrix Say that the diagonal entries are a1 an The hypothesis is that a2i 0 Since the ai are real numbers they all must be 0 Hence A 0 5 Let T V W be a linear transformation between finite dimensional vector spaces Let X be a subspace of W Let T 1 X be the set of vectors in V that map to X Show that T 1 X is a subspace of V and that dim T 1 X dim V dim W dim X This seems to be problem 2 of the further review problems As I write this answer I have the impression that the problem is harder than I thought but perhaps there s an easier way to say what I m about to explain Let Y be the range of T so that X and Y are both subspaces of W A third subspace is X Y Consider the quotient space W X and the natural map Y W X that sends y Y to y X The null space of this map is Y X Hence dim Y dim Y X rank dim Y X dim W X dim Y X dim W dim X Now let U be the restriction of T to T 1 X Since T 1 X contains the null space of T the nullity of U is the same thing as the nullity of T The range of U is Y X We have dim T 1 X nullity T dim Y X nullity T dim X dim Y dim W dim V dim X dim W where we have used the equality dim V nullity T rank T nullity T dim Y 6 Suppose that V is a real finite dimensional inner product space and that T V V is a linear transformation with the property that hT x T y i 0 H110 final page 2 whenever x and y are elements of V such that hx yi 0 Assume that there is a non zero v V for which kT v k kvk Show that T is orthogonal This is a slightly friendlier version of problem 8 on HW 14 After scaling v we may assume that kvk 1 Complete v to a basis of V and then apply the GramSchmidt process We emerge with an orthonormal basis e1 en of V with v e1 Let i be greater than 1 and let w ei Then hv w v wi hv vi hw wi 2 because v w Similarly hT v w T v w i kT v k2 kT w k2 because T v T w by hypothesis It follows that kT w k2 1 because we knew already that kT v k2 was 1 In other words we have kT ei k 1 for all i equivalently hT ei T ej i ij for all i and j It follows by linearity that hT x T y i hx yi for x y V Thus T is orthogonal 7 Let T be a nilpotent operator on a …