# Euclidean and Unitary Spaces

• Nathan Jacobson
Part of the Graduate Texts in Mathematics book series (GTM, volume 31)

## Abstract

As we have pointed out at the beginning of these Lectures, Euclidean geometry is concerned with the study of a real vector space relative to the scalar product $$\sum\limits_1^n {{\xi _i}{\eta _i}}$$ determined by writing x Σξ i u i y = Ση i u i in terms of basic unit vectors that are mutually orthogonal. Since this scalar product is fixed, it is customary to denote it simply as (x, y) instead of g(x, y) as in the preceding chapter. The geometric meaning of (x, y) is clear. It gives the product of the cosine of the angle between x and y by the lengths of the two vectors. The length of x can also be expressed in terms of the scalar product, namely, |x| = (x,x)½

## Keywords

Scalar Product Linear Transformation Scalar Multiplication Orthogonal Matrix Euclidean Geometry
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