Appendix A — Homework 0

Exercise A.1 Let \(V\) and \(W\) be \(F\)-vector spaces and let \(L: V \to W\) be a linear map. Take \(x \in V\) and set \(y = L(x) \in W\). Prove that \[ L^{-1}(y) = x + \ker L. \]

Exercise A.2 Let \(F\) be a field and suppose that \(\phi: \operatorname{Mat}_n(F) \to F\) satisfies \(\phi(AB) = \phi(A) \phi(B)\) for all \(A, B \in \operatorname{Mat}_n(F)\) and that \(\phi(\mathbb{I}) = 1\).

  1. If \(A \in \operatorname{GL}_n(F)\), show that \(\phi(A^{-1}) = \phi(A)^{-1}\).

  2. Suppose henceforth that \(\phi\) is non-trivial, i.e., that \(\phi\) is not the constant map sending every matrix to \(1\). Show \(\phi(0) = 0\).

  3. Show that that \(\phi(J) = 0\), where \(J\) is the matrix with \(1\)s along the superdiagonal and \(0\)s elsewhere.

    Hint: What is \(J^2\)? What about other powers of \(J\)?

  4. Note that if the columns of \(M \in \operatorname{Mat}_n(F)\) are \(c_1, c_2, \dots, c_n\), then the columns of \(MJ\) are \(0, c_1, \dots, c_{n-1}\). Now, suppose \(\operatorname{rank}(A) = n-1\). Show that there is always a basis change \(S \in \operatorname{GL}_n(F)\) such that the first column of \(S^{-1} A S\) is zero. Conclude that \(A = T J S^{-1}\) for some \(S, T \in \operatorname{GL}_n(F)\) and hence \(\phi(A) = 0\).

Since any non-invertible matrix \(A\) may be written as a product of rank \(n-1\) matrices, we can conclude that \(\phi(A) = 0\) for all non-invertible \(A \in \operatorname{Mat}_n(F)\) and hence \(\phi\) is determined by its values on \(\operatorname{GL}_n(F)\).

Exercise A.3 Show \(\operatorname{Tr}(AB) = \operatorname{Tr}(BA)\) for all \(A, B \in \operatorname{Mat}_n(F)\); conclude that the trace is independent of the choice of basis.

Exercise A.4 Show that any linear map \(f: \operatorname{Mat}_n(F) \to F\) satisfying \(f(AB) = f(BA)\) for all \(A, B \in \operatorname{Mat}_n(F)\) and \(f(\mathbb{I}) = n\) is the trace.

Hint: Consider the standard basis of matrices, i.e., those of the form \(E_{i,j} \in \operatorname{Mat}_n(F)\) for all \(1 \leq i, j \leq n\). What happens when you multiply these matrices?

Exercise A.5 Let \(\lambda \in \mathbb{C}\setminus \{0\}\). Show that the Heisenberg relation \(AB - BA = \lambda \mathbb{I}\) cannot hold for matrices, i.e., quantum mechanics requires infinite dimensions.

Hint: Take the trace of both sides.

Exercise A.6 A matrix \(S \in \operatorname{Mat}_n(F)\) is called a permutation matrix if it has exactly one entry of \(1\) in each row and each column and \(0\) elsewhere—such a matrix induces a permutation \(\sigma \in \mathcal{S}_{n}\) through its action on column vectors. For example \[ (1\ 2\ 4) \in \mathcal{S}_{4} \mapsto \begin{pmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}. \]

Recall that \(i\) is a fixed point of \(\sigma \in \mathcal{S}_{n}\) if \(\sigma(i) = i\). Show that \[ \left|\operatorname{Fix}(\sigma)\right| = \operatorname{Tr}(S). \]

Exercise A.7 Describe all group homomorphisms \(\mathbb{Z}/{n}\mathbb{Z} \to \mathbb{C}^\times\).