文档介绍：PROBLEMS AND THEOREMS
IN LINEAR ALGEBRA
V. Prasolov
Abstract. This book contains the basics of linear algebra with an emphasis on non-
standard and neat proofs of known theorems. Many of the theorems of linear algebra
obtained mainly during the past 30 years are usually ignored in text-books but are
quite accessible for students majoring or minoring in mathematics. These theorems
are given plete proofs. There are about 230 problems with solutions.
Typeset by AMS-TEX
1
CONTENTS
Preface
Main notations and conventions
Chapter I. Determinants
Historical remarks: Leibniz and Seki Kova. Cramer, L’Hospital,
Cauchy and Jacobi
1. Basic properties of determinants
The Vandermonde determinant and its application. The Cauchy deter-
minant. Continued fractions and the determinant of a tridiagonal matrix.
Certain other determinants.
Problems
2. Minors and cofactors
-Cauchy’s formula. Laplace’s theorem. Jacobi’s theorem on minors
of the adjoint Theř generalized Sylvester’s identity. Chebotarev’s
ř ij řp−1
theorem on the matrix ε 1 , where ε= exp(2πi/p).
Problems
3. The plementű
A11 A12 −1
Given A = , the matrix (A|A11) = A22 − A21A11 A12 is
A21 A22
called the plement (of A11 in A).
. det A = det A11 det (A|A11).
. Theorem. (A|B) = ((A|C)|(B|C)).
Problems
k k
4. Symmetric functions, sums x1 +· · ·+xn, and Bernoulli numbers
Determinant relations between σ(x , . . . , x ), s (x , . . . , x ) = xk +· · ·+
P k 1 n k 1 n 1
k i1 in
xn and pk(x1, . . . , xn) = x1 . . . xn . A determinant formula for
i1+...ik=n
n n
Sn(k) = 1 + · · · + (k − 1) . The Bernoulli numbers and Sn(k).
. Theorem. Let u = S1(x) and v = S2(x). Then for k ≥ 1 there exist
2
polynomials pk and qk such that S2k+1(x) = u pk(u) and S2k(x) = vqk(u).
Problems
Solutions
Chapter II. Linear spaces
Historical remarks: Hamilton and Grassmann
5. The dual space. The plement
Linear equations and their application to the following theorem:
5.