Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.1

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.1

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8.1 Complex Numbers 8.2 Conjugates and Division of Complex Numbers 8.3 Polar Form and DeMoivre’s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5 Unitary and Hermitian Matrices COMPLEX VECTOR SPACES 8 469 harles Hermite was born on Christmas Eve in Dieuze, France––the sixth of seven children. His father was a cloth merchant who had studied engineering. Although much of his early education occurred at home, he went to Paris to study when he was eighteen. At the age of 20, Hermite published a paper titled Considerations on the Algebraic Solution of the Equation of the Fifth Degree. (Later in his life, Hermite showed how to solve a general fifth-degree equation by means of elliptic modular functions––such equations cannot be solved generally by algebraic means.) That same year, Hermite was admitted to the École Polytechnique, but he was dismissed from the school after one year. While at the Polytechnique, Hermite became acquainted with Joseph Liouville, who in turn introduced Hermite to Carl Jacobi. Hermite’s correspondence with Jacobi reveals Hermite’s early understanding of abstract mathematics. In 1848, Hermite gained a position at the Polytechnique (the same institution that had dismissed him five years earlier). With this ap- pointment, Hermite’s career finally began to take shape. In 1856 he was appointed to the Academy of Sciences, in 1869 to the École Normale, and finally in 1870 to a professor- ship at the Sorbonne, a position he held until his retirement in 1897. Hermite is best known for his work with elliptic functions, for his proof that e is a tran- scendental number, and for his introduction of what are today called Hermite polynomials. Hermite was the first to use the term orthogo- nal matrices , and was the first to prove that if a matrix is equal to its own conjugate transpose, then its eigenvalues must be real. 8.1 COMPLEX NUMBERS Thus far in the text, the scalar quantities used have been real numbers . In this chapter, you will expand the set of scalars to include complex numbers. In algebra it is often needed to solve quadratic equations such as The general quadratic equation is and its solutions are where the quantity in the radical, is called the discriminant. If then the solutions are ordinary real numbers. But what can you conclude about the solutions of a quadratic equation whose discriminant is negative? For example, the equation b 2 2 4 ac 0, b 2 2 4 ac , x 5 2 b 6 ! b 2 2 4 ac 2 a ax 2 1 bx 1 c 5 0 x 2 2 3 x 1 2 5 0. Charles Hermite 1822–1901 C

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has a discriminant of From your experience with ordinary algebra, it is clear that there is no real number whose square is However, by writing you can see that the essence of the problem is that there is no real number whose square is To solve the problem, mathematicians invented the imaginary unit i , which has the property that In terms of this imaginary unit, you can write The imaginary unit i is defined as follows.
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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 8.1

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