Advanced Engineering Mathematics, 7th Edition

7.6 Vector Spaces

INTRODUCTION

In the preceding sections we were dealing with points and vectors in 2- and 3-space. Mathematicians in the nineteenth century, notably the English mathematicians Arthur Cayley (1821–1895) and James Joseph Sylvester (1814–1897) and the Irish mathematician William Rowan Hamilton (1805–1865), realized that the concepts of point and vector could be generalized. A realization developed that vectors could be described, or defined, by analytic rather than geometric properties. This was a truly significant breakthrough in the history of mathematics. There is no need to stop with three dimensions; ordered quadruples 〈a₁, a₂, a₃, a₄〉, quintuples 〈a₁, a₂, a₃, a₄, a₅〉, and n-tuples 〈a₁, a₂, … , a_n〉 of real numbers can be ...