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CHAPTER

Euclidean Vector Spaces

I N T R O D U C T I O N : The idea of using pairs of numbers to locate points in the plane and triples of numbers to locate points

in 3-space was first clearly spelled out in the mid-seventeenth century. By the latter part of the eighteenth century,

mathematicians and physicists began to realize that there was no need to stop with triples. It was recognized that quadruples

of numbers could be regarded as points in “four-dimensional” space, quintuples as points

in “five-dimensional” space, and so on, an n-tuple of numbers being a point in “ n-dimensional” space. Our goal in this chapter
is to study the properties of operations on vectors in this kind of space.

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