# Math 817: Problem set 7

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, October 15, 2004. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)
1. First, some prologue: An mxn matrix A with entries in a field F gives a linear transformation TA: Fn -> Fm, by matrix multiplication. The rank of A is defined to be the rank of TA. (Since rank(TA) = dim Im(TA), we see that the rank of A is just the dimension of the column space of A, which is given by counting the number of pivots in the row reduced matrix for A, which itself is just the matrix for TA, but with respect to a different basis for Fm.) Now the problem. Show that A and At, the transpose of A (p. 18 of Artin), have the same rank.
2. Let F be a field and let W be a subspace of Fm. Recall from the previous homework the subspace W'.
• (a) Prove that dim W + dim W' = dim Fm.
• (b) Prove that W = (W')'.
• (c) Let i be an integer between 0 and m, inclusive. Prove that there is a bijection between subspaces of Fm of dimension i and those of dimension m-i.
3. Let A be an mxn matrix with entries in a field F, and let TA: Fn -> Fm be the corresponding linear transformation. Let V be a subspace of Fn, and let W be a subspace of Fm such that TA(V) is contained in W. Let T' be the linear transformation corresponding to At. Show that T'(W') is contained in V', where the operation ' is defined as on the previous homework.
4. Count the number of subspaces of each dimension in F3 if F is the field F5. Justify your answers.
• (a) Find the rank of the matrix given in problem 1.2 of Artin, p. 145.
• (b) When F is a finite field with p elements (p a prime), formula 1.6 (p. 110) and formula 6.15 (p. 58) both apply to a linear transformation of finite dimensional vector spaces over F. Show how each formula implies the other. (Once you understand what to do, this one is easy!)
5. (Problem 2.9 on p. 146 of Artin.) Let S and T be linear transformations from V to W. Define S+T and cT by the rules [S+T](v) = S(v) + T(v) and [cT](v) = c(T(v)).
• (a) Prove that S+T and cT are linear transformations, and describe their matrices in terms of the matrices for S and T.
• (b) Let L be the set of all linear transformations from V to W. Prove that these laws make L into a vector space, and compute its dimension.
6. (Problem 3.5 on p. 146 of Artin.) Find all of the invariant subspaces of the real linear operators TA and TB whose associated matrices are (a) A and (b) B, where A is 2x2, has a 0 at bottom left, and otherwise its entries are 1s, and B is a 3x3 diagonal matrix with 1, 2 and 3 on the diagonal. (I.e., find each invariant subspace, and justify why there are no others.)