Unit 4 of 10
Inner Product Spaces
- 6 lessons
- Videos 16 to 21
This unit sets up inner products over the real and complex numbers and uses them to measure length, project onto subspaces and find best approximations.
Conjugation keeps the squared length of every nonzero vector positive. With that in place come the Cauchy-Schwarz inequality and its equality case, orthonormal expansions built by the complex Gram-Schmidt process, Parseval's identity and Bessel's inequality. The Riesz representation theorem matches each linear functional with a vector, which leads to the adjoint of a map. The unit ends with the projection theorem and weighted least squares, where changing the inner product can change the best fit.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Complex Geometry and Orthonormal Coefficients
Complex Inner Products
Why complex length needs the conjugate transpose, the complex inner product with conjugate symmetry and positive definiteness, sesquilinear forms, the norm it defines and the Gram matrix of a basis.
Video 16
Cauchy-Schwarz and Orthonormal Expansions
How subtracting a projection proves the Cauchy-Schwarz inequality and its equality case and leads to the triangle inequality, the complex Gram-Schmidt process, orthonormal expansions, Parseval's identity and Bessel's inequality.
Video 17
Representation and Adjoints
Riesz Representation in Finite Dimensions
The Riesz representation theorem in finite dimensions, the representing vector of a functional, its construction from an orthonormal basis, the conjugate-linear identification and how the vector depends on the inner product.
Video 18
Adjoints and Gram Matrices
How the adjoint of a linear map comes from Riesz representation, why adjoints reverse compositions, how Gram matrices enter the adjoint's matrix, and how orthogonal complements relate its kernel and image to those of the map.
Video 19
Projection and Best Approximation
Orthogonal Projection and Best Approximation
The orthogonal decomposition of an inner product space, the orthogonal projection and the projection theorem for best approximation, orthogonal projectors as self-adjoint idempotent maps, and polynomial approximation.
Video 20
Weighted Least Squares
How a positive definite weight matrix gives a weighted inner product, weighted orthogonality of residuals, the weighted normal equations, why the fit is unique and when its coefficients are, and inner products from sample values.
Video 21