- First, some prologue: An mxn matrix A with entries in a field F gives a
linear transformation T
_{A}: F^{n}-> F^{m}, by matrix multiplication. The rank of A is defined to be the rank of T_{A}. (Since rank(T_{A}) = dim Im(T_{A}), 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 T_{A}, but with respect to a different basis for F^{m}.) Now the problem. Show that A and A^{t}, the transpose of A (p. 18 of Artin), have the same rank. - Let F be a field and let W be a subspace of F
^{m}. Recall from the previous homework the subspace W'.- (a) Prove that dim W + dim W' = dim F
^{m}. - (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 F
^{m}of dimension i and those of dimension m-i.

- (a) Prove that dim W + dim W' = dim F
- Let A be an mxn matrix with entries in a field F,
and let T
_{A}: F^{n}-> F^{m}be the corresponding linear transformation. Let V be a subspace of F^{n}, and let W be a subspace of F^{m}such that T_{A}(V) is contained in W. Let T' be the linear transformation corresponding to A^{t}. Show that T'(W') is contained in V', where the operation ' is defined as on the previous homework. - Count the number of subspaces of each dimension in F
^{3}if F is the field**F**_{5}. 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!)

- (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.

- (Problem 3.5 on p. 146 of Artin.) Find all of the invariant
subspaces of the real linear operators T
_{A}and T_{B}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.)