Inner Product Spaces
Stub. Inner products and induced norms, the Gram-Schmidt orthogonalization process and orthonormal bases, orthogonal complements, minimization problems, and the pseudoinverse.
Adds geometry (length, angle, orthogonality) to a vector space via an inner product. Gram–Schmidt turns any basis into an orthonormal one, which makes orthogonal complements and the associated minimization problems (closest point in a subspace, least-squares) tractable — the linear-algebra core of the pseudoinverse, ahead of its full treatment via the SVD.
Outline (TODO — flesh out each)
- Inner products and norms
- The Gram–Schmidt orthogonalization process; orthonormal bases
- Orthogonal complements
- Minimization problems (least-squares); the pseudoinverse (intro)
- Linear functionals on inner product spaces