• Răzvan Gelca
  • Titu Andreescu


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 operations, groups, and rings, really no further than the definition of a group or a ring.


Triangle Inequality Positive Real Number Binary Operation Identity Element Chebyshev Polynomial 
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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Răzvan Gelca
    • 1
  • Titu Andreescu
    • 2
  1. 1.Department of Mathematics and StatisticsTexas Tech UniversityLubbockUSA
  2. 2.School of Natural Sciences and MathematicsUniversity of Texas at DallasRichardsonUSA

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