Math 551, Assignment 7, due Monday, December 2, in my mailbox

Throughout this assignment, V is a finite-dimensional vector space over the field k, and B = {v_{1}, …, v_{n}} is an ordered basis of V.

1. Let B be a bilinear form on V. Let β_{L}, β_{R} be the associated mappings from V to V^{∗}. Show that

a) β_{L} = β_{R}^{∗}

b) [β_{R}]_{B^{∗}}^{B} = {B}_{B}. (Or is it [β_{L}]_{B^{∗}}^{B}?)

c) The set of all matrices of the form {B}_{B}, as B varies over all the ordered bases of V (but B remains fixed) is an equivalence class under the equivalence relation ≈ defined by A ≈ A′ if and only if A′ = P^{T} AP for some invertible P.

2. Suppose that T : V → V is a diagonalizable linear transformation. Let W ⊆ V be a subspace such that T (W) ⊆ W. Thus T induces linear transformations T |W : W → W and T |^{V/W} : V/W → V/W, which you are asked to prove are both diagonalizable as well. (Hint. Relate their minimal polynomials to μ_{T}.)

3. Suppose that T : V → V and U : V → V are linear transformations such that

T U = UT.

a) Show that for any eigenspace W of T, or indeed for any generalized eigenspace W of T, U(W) ⊆ W.

b) Suppose that T and U are both diagonalizable (that is, there is a basis of V whose elements are eigenvectors for T, and there is also a basis whose elements are eigenvectors for U). Show that they are simultaneously diagonalizable, that is, there is a basis of V whose elements are eigenvectors both for T and for U. (Use the previous problem.)

c) Interpret b) as a theorem about matrices.

4. A lattice Λ in R^{n} is defined to be an abelian subgroup of R^{n} (under addition) which is generated by an R-basis B of R^{n}.

a) Show that Λ is then a free abelian group with basis B = {v_{1}, …, v_{n}}.

The lattice Λ is said to be integral if and only if (v, w) ∈ Z for all v, w ∈ Λ. Here ( ·, ·) is the standard Euclidean inner product. The lattice dual of Λ is defined to be the group

Λ^{∗} = {v ∈ R^{n} | (v, w) ∈ Z for all w ∈ Λ}.

b) If Λ is integral, show that Λ ⊆ Λ^{∗}, that

|Λ^{∗}/Λ| = | det[(v_{i}, v_{j})]_{i,j=1}^{n}|,

and that the structure of Λ^{∗}/Λ can be revealed by applying suitable integer row and column operations to the matrix [(v_{i}, v_{j})]_{i,j=1}^{n}.

(Hint. Let w_{1}, …, w_{n} ∈ R^{n} be such that (v_{i}, w_{j}) = δ_{ij} (why do these vectors exist?) and show that Λ^{∗} is a lattice with basis {w_{1}, …, w_{n}}. Then see the previous assignment.)

c) What is the structure of the group Λ^{∗}/Λ in the following cases?

(i) B = { [ 1 −1 0 ], [ 0 1 −1 ] }

(ii) B = { [ 1 −1 0 0 ], [ 0 1 −1 0 ], [ 0 0 1 −1 ], [ 0 0 1 1 ] }.