Answers to some of homework is here; I willl continue to add to it.

Homework #1 (due Friday Sep 11):

  • Read sections 1.1-1.3 of the book
  • Exercises 1.1, 1.4, 1.8
  • Homework #2 (due Friday Sep 18, answers here):
  • Compute the derivative of the determinant map M_n(R) -> R.
  • Compute the critical points (in the source) and critical values (in the target) of the map f : M |-> M M^T, from M_n(R) -> symmetric matrices.
  • Let O(n) denote the set of orthogonal matrices, i.e. f^{-1}(identity).
    Compute the tangent space to each point of O(n).
  • Let G be an nxn diagonal matrix with k 1s and n-k 0s on the diagonal. (EDIT: you may take them consecutive -- all 1s, then all 0s.) Let h : O(n) -> symmetric matrices take M |-> M G M^T.
    Compute its derivative and the rank of the derivative.
    HINT: it's easier to compute the perpendicular space to the image than the image itself. Two matrices A,B are perpendicular if trace(A B^T) = 0.
  • Homework #3 (due Friday Sep 25):
  • Let M be an invertible nxn matrix, i.e. its columns v_1..v_n are a basis of R^n. Let N be the matrix whose rows are the dual basis. How succinctly can you express this dual-basis condition, as a matrix equation involving M and N? (Hint: very.)
  • Let V,W be vector spaces, and let v in V, f in W^* be vectors. Define f@v : W -> V as the linear map taking w |-> f(w)v.
  • Say M : W -> V is a linear map. How do you test whether M is of the form f@v for some pair f,v?
  • If V=W, then f(v) means something (it's a number). If V=W=R^n, then M : R^n -> R^n i.e. we can write it as a matrix. How do you compute f(v) in terms of the matrix M?
  • Let f : R^2 -> R^2 take (x,y) |-> (e^x, xy). Compute f^* of the following forms:
  • alpha(x,y) = x+y, a 0-form
  • alpha(x,y) = y dx, a 1-form
  • alpha(x,y) = xy dx ^ dy, a 2-form
  • Homework #4 (due Friday Oct 2): book problems 2.1, 2.5, 2.10, 2.11.

    Homework #5 (due Friday Oct 9): book problems 2.9, 3.6, 3.12, 3.16

    Homework #6 (due Friday Oct 16):

  • Let M be a diagonalizable matrix, i.e. one with a basis v_1...v_n of eigenvectors. Show that the dual basis elements are also eigenvectors, of M^*.
  • book problems 3.15, 4.1
  • Homework #7 (due Friday Oct 23):

  • book problems 5.1, 5.2, 5.3
  • Let b: R^n x R^m -> R be a bilinear map.
  • Show that there exists a unique nxm matrix M such that b(v,w) = trace(M w v^T) for all column vectors v,w.
  • Describe the two subspaces of R^n, R^m that we have to quotient by in order to make this a perfect pairing, in terms of M.
  • Homework #8 (due Friday Nov 6):
  • Let M = R^2 \ 0, the punctured plane. We computed its ordinary cohomology H^*(M) a few weeks ago.
  • Compute its compactly supported cohomology H^*_c(M), i.e. using the d operator on compactly supported forms. (Careful: the set of vectors v in R^2 with |v| at most r is compact, but once you remove 0 it's not compact.)
  • Show that each pairing H^i_c(M) x H^{2-i}(M) -> R, by wedging then integrating, is perfect.
  • book problem 8.6
  • Given orientations on a pair V < W of vector spaces, show how to define an orientation on the quotient space W/V. Your rule should have the property that if you flip V or W the rule flips W/V.
  • Homework #9 (due Friday Nov 13):
  • book problems 9.1, 9.2 (recall the book writes ab for a ^ b), 9.5
  • Homework #10 (due Friday Nov 20):
  • Let T: A->B, U: C->B. Break up A,B,C as direct sums in ways that T,U respect, such that on each piece T,U are either 0 or isomorphisms.
  • Put in another space V: D->B, and assume B is 2-dimensional while A,C,D are 1-dimensional. Describe all the possibilities, up to change of basis in A,B,C,D.
  • Homework #11 (due Friday Dec 4):
  • Let M be a manifold. Define the PoincarĂ© polynomial of M as p_M(t) := sum_{i=0}^{dim M} (dim H^i(M)) t^i, and the Euler characteristic as Chi(M) := p_M(-1).

    Use Mayer-Vietoris to show that if M = U union V (open submanifolds), then Chi(M) = Chi(U) + Chi(V) - Chi(U intersect V).

  • Compute the Euler characteristic of a genus g surface using this.
  • What's the best statement you can make like that formula, relating the PoincarĂ© polynomials?