Mathematics 505        Winter 2004

Homework

• Due Wednesday, January 14:
  • Section 10.1: 7, 9, 13, 19
  • Given a vector space V, a vector space homomorphism T from V to itself, and a basis B for V, let A be the matrix for T with respect to B. Show that finding all bases for which the matrix of T equals A is equivalent to finding all invertible matrices which commute with A.
  • For the vector space R² and the linear operator T on R² defined by T(a,b) = (2a+b, 3a-b), find the matrix A for T with respect to the standard basis, and find all bases for R² with respect to which the matrix for T equals A.
  • (Section 11.1: 10-14 are interesting, but you don't need to turn them in.)
• Due Wednesday, January 21:
  • Section 10.1: 5
  • Section 10.2: 4, 6, 7, 9, 10, 13
• Due Wednesday, January 28:
  • Section 10.3: 7, 9, 11, 12, 25 (problems 2, 24, and 26 are worth noting, also)
  • Show that an R-module M is cyclic if and only if M is isomorphic to R/I for some left ideal I of R.
• Due Wednesday, February 4:
  • Section 10.4: 11, 14, 15
  • Section 10.5: 2, 3, 4, 8
  • Comments on other problems from 10.5: You might also do problem 1 on your own, if you need more practice with this sort of thing. Problems 12 and 13 are worth noting; problem 12 could be taken as the definition of direct sums and direct products of R-modules. You can use problem 12 to study the analogues of problems 3 and 4 for infinite direct sums and products; you can show that arbitrary sums of projective modules are projective, and arbitrary products (but not sums) of injective modules are injective.
• No homework due Wednesday, February 11. Practice problems (on the most recent material) for the exam:
  • Section 10.5: 6, 7, 9, 10, 21, 22, 28 (15-17 are worth noting, since they give a proof that every module is contained in an injective module. Also, 26 gives a classification of flat modules over a PID.)
• Due Wednesday, February 18:
  • Section 11.3: 1, 2, 3, 4
  • Let V and W be vector spaces over a field F. Find a one-to-one linear transformation from (V^* tensor W) to Hom_F(V,W). Conclude that if V and W are finite-dimensional, this map is an isomorphism.

    (For example, (V^* tensor V) is isomorphic to Hom(V, V), which is in turn isomorphic to the space of n x n matrices, if dim V = n.)

    (You can also use this with tensor-hom adjointness -- Theorem 43 in Chapter 10 -- to get a nice isomorphism between (V tensor W)^* and (V^* tensor W^*), if V and W are finite-dimensional.)

• Due Wednesday, February 25:
  • Let V be a finite-dimensional vector space over a field F. By one of last week's problems, there is an isomorphism between (V^* tensor V) and Hom(V,V). The evaluation map (V^* tensor V) -> F, which takes (f tensor v) to f(v), corresponds to a map Hom(V,V) -> F. What map is it? (That is, if I view it as a map from the space of n x n matrices to the field, where does it send a matrix A?)
  • Section 11.5: 1, 6, 12, 13, 14
• Due Wednesday, March 3:
  • Section 12.1: 2, 3, 11
  • Let M be a module over an integral domain R. Show that M is a torsion module if and only if the rank of M is 0.
  • Let Z be the integers, and let N be the submodule of Z³ generated by the elements (1, 2, 3), (4, 5, 6), and (7, 8, 9). Write Z³ / N as a direct sum of cyclic groups. (This was a prelim problem a few years ago.)
  • Let k be a field, and classify the finitely generated modules over R = k[x] / (x³). (Warning: R is not a PID.)
  • You may use problem 19 and the discussion surrounding it to do any of these, if you want; it's a worthwhile problem to look at in any case. Problems 21 and 22 also give nice applications of the classification theorems of this section.
• Due Wednesday, March 10:
  • Section 12.2: 4, 6 (you can use Jordan form to do this problem, if you want), 11, 12 (just do the first part), 15, 17
• Practice problems on Jordan form:
  • Section 12.3: 2, 10, 17, 18, 20, 21, 23, 24, 25, 26
  • Show that if A and B are complex n x n matrices, then they are simultaneously diagonalizable if and only they are each diagonalizable, and they commute. (More generally, you can work with any set of n x n matrices, and over any field which contains all of their eigenvalues.)

       John Palmieri    Padelford C-538    (206) 543-1785    email    web page