2 Algebra........................................................
It is now time to split mathematics into branches. First, algebra. A section on algebraic
identities hones computational skills. It is followed naturally by inequalities. In general,
any inequality can be reduced to the problem of finding the minimum of a function.
But this is a highly nontrivial matter, and that is what makes the subject exciting. We
discuss the fact that squares are nonnegative, the Cauchy–Schwarz inequality, the triangle
inequality, the arithmetic mean–geometric mean inequality, and also Sturm’s method for
proving inequalities.
Our treatment of algebra continues with polynomials. We focus on the relations
between zeros and coefficients, the properties of the derivative of a polynomial, problems
about the location of the zeros in the complex plane or on the real axis, and methods for
proving irreducibility of polynomials(such as the Eisenstein criterion). From all special
polynomials we present the most important, the Chebyshev polynomials.
Linear algebra comes next. The first three sections, about operations with matrices,
determinants, and the inverse of a matrix, insist on both the array structure of a matrix
and the ring structure of the set of matrices. They are more elementary, as is the section on
linear systems. The last three sections, about vector spaces and linear transformations,
are more advanced, covering among other things the Cayley–Hamilton Theorem and the
Perron–Frobenius Theorem.
The chapter concludes with a brief incursion into abstract algebra: binary opera-
tions, groups, and rings, really no further than the definition of a group or a ring.
2.1 Identities and Inequalities
2.1.1 Algebraic Identities.......................................
The scope of this section is to train algebraic skills. Our idea is to hide behind each
problem an important algebraic identity. We commence with three examples, the first