Math 55a: Honors Abstract Algebra
Homework Assignment #9 (1 November 2010):
Linear Algebra IX: Exterior algebra and determinants, cont’d
*, 231
—Axler, Symbol Index on p.247
The symbols *that appear on the first page of these chapters are decorations
[. .. ] Chapter 1 has one of these symbols, Chapter 2 has two of them, and so on.
The symbols get smaller with each chapter. What you may not have noticed
is that the sum of the areas of the symbols at the beginning of each chapter
is the same [. . . ]
—Axler, p.231 [answering a question posed in PS1]
A couple of Axler problems to start with:
1. i) Solve Problem 9 on page 244 of the textbook. What is the trace of P?
ii) Solve Problem 22 on page 246. This should be easy to do with the Vndefinition of
the determinant, even though Axler didn’t intend that.
Symmetric pairings from exterior algebra:
2. Let Vbe a vector space of dimension 4 over any field Fnot of characteristic 2.1Let
W=V2V, and fix a nonzero ψ∈V4V. We have seen that there is a symmetric
nondegenerate pairing W×W→Fdefined by ω∧ω0=hω, ω0iψ. Prove that
hω, ωi= 0 if and only if ω=v1∧v2for some v1, v2∈V.
3. Now let Vbe a vector space of dimension 4nover Rfor some positive integer n. Let
W=V2nV, and fix a nonzero ψ∈V4nV. Again there is a symmetric nondegen-
erate pairing W×W→Fdefined by ω∧ω0=hω, ω0iψ. Since Wis now a real
vector space, this pairing has a signature. Find it, and construct a decomposition
W=W+⊕W−into orthogonal subspaces such that the pairing is positive-definite
on W+and negative-definite on W−.
The determinant and other properties of a circulant matrix:
4. An n×nmatrix Ais said to be circulant when its (i, j)-th entry aij depends only on
i−jmod n. Let Sbe the circulant matrix for which aij is 1 if j≡i+1 mod nand 0
otherwise. Show that Ais circulant if and only if Ais a polynomial in S. Conclude
that all circulant matrices commute and are normal. What are the eigenvalues,
eigenvectors, and determinant of a circulant matrix with entries in C? [This can be
viewed as part of a theory of discrete Fourier analysis, or of representations of the
cyclic group Z/nZ. The case n= 2 of the determinant formula is well known; less
so the n= 3 determinant, though it is still memorable, and shows up every once in
a while — usually in contexts less frivolous than my observation about 27 years ago
that 1983 |233 −5·217 −1.]
Linear algebra as a tool for studying field algebra, continued: Let Fbe any field, Pthe
polynomial ring F[z], and Pk⊂ P (k= 0,1,2, . . .) the F-vector space of polynomials of
degree at most k.
1The construction here can be modified to work in characteristic 2 as well; the result stays the same.
