Module 7: Inner Product Spaces
Explore the geometry of vector spaces through dot products, orthogonality, projections, and the Gram-Schmidt process, culminating in least-squares applications.
Module Lessons
1
Dot Product, Length, and Orthogonality
Define the dot product and its properties, compute vector lengths and distances, and understand orthogonal sets.
2
Orthogonal Projections
Project vectors onto lines and subspaces, and decompose vectors using the orthogonal decomposition theorem.
3
The Gram-Schmidt Process
Convert any basis into an orthogonal or orthonormal basis using the Gram-Schmidt algorithm.
4
Least Squares Problems
Solve overdetermined systems using the normal equations and apply least squares to data fitting.