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Linear Algebra pp 244-273 | Cite as

Inner product spaces

  • Larry Smith
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

So far in our study of vector spaces and linear transformations we have made no use of the notions of length and angle, although these concepts play an important role in our intuition for the vector algebra of ℝ2 and ℝ3. In fact the length of a vector and the angle between two vectors play very important parts in the further development of linear algebra and it is now time to introduce these ingredients into our study. There are many ways to do this and in the approach that we will follow both length and angle will be derived from a more fundamental concept called a scalar or inner product of two vectors. No doubt the student has encountered the scalar product in the guise of the dot product of two vectors in ℝ3 which is usually defined by the equation
$$ {\rm{A}} \cdot {\rm{B = }}\left| {\rm{A}} \right|\left| {\rm{B}} \right|{\rm{ cos }}\theta $$
where |A| is the length of the vector A, similarly for B, and θ is the angle between A and B. In the study of vectors in ℝ3 this is a reasonable way to introduce the scalar product, because lengths and angles are already defined and well-studied concepts of geometry. In a more abstract study of linear algebra, such as we are undertaking, such an approach is not possible, for what is the length of a polynomial (vector) in \( {P_4}(R) \)? This problem can be overcome by the use of the axiomatic method. Having introduced vector spaces by axioms it is not at all unreasonable to employ additional axioms to impose further structure on them.

Keywords

Vector Space Scalar Product Linear Transformation Linear Subspace Product Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1984

Authors and Affiliations

  • Larry Smith
    • 1
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenWest Germany

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