9: Inner product spaces
- Page ID
- 257
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The abstract definition of a vector space only takes into account algebraic properties for the addition and scalar multiplication of vectors. For vectors in \(\mathbb{R}^n\), for example, we also have geometric intuition involving the length of a vector or the angle formed by two vectors. In this chapter we discuss inner product spaces, which are vector spaces with an inner product defined upon them. Inner products are what allow us to abstract notions such as the length of a vector. We will also abstract the concept of angle via a condition called orthogonality.
- 9.5: The Gram-Schmidt Orthogonalization procedure
- We now come to a fundamentally important algorithm, which is called the Gram-Schmidt orthogonalization procedure. This algorithm makes it possible to construct, for each list of linearly independent vectors (resp. basis), a corresponding orthonormal list (resp. orthonormal basis).
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.