Lectures in Abstract Algebra pp 172-198 | Cite as

# Euclidean and Unitary Spaces

## 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## Preview

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