Linear Algebra and Geometry 3
Inner product spaces, quadratic forms, and more advanced problem solving
C1 Eigendecomposition, spectral decomposition
1. Introduction to the course
2. Geometrical operators in the plane and in the 3-space
You will learn: using eigenvalues and eigenvectors of geometrical operators such as symmetries, projections, and rotations in order to get their standard matrices; you will also strengthen your understanding of geometrical transformations.
3. More problem solving; spaces different from R^n
You will learn: work with eigendecomposition of matrices for linear operators on various vector spaces.
4. Intermezzo: isomorphic vector spaces
You will learn: about certain similarities between different spaces and how to measure them.
5. Recurrence relations, dynamical systems, Markov matrices
You will learn: more exciting applications of eigenvalues and diagonalization.
6. Solving systems of linear ODE, and solving higher order ODE
You will learn: solve systems of linear ODE and linear ODE of higher order with help of diagonalization.
C2 Inner product spaces
7. Inner product as a generalization of dot product
You will learn: about other products with similar properties as dot product, and how they can look in different vector spaces.
8. Norm, distance, angles, and orthogonality in inner product spaces
You will learn: how to define geometric concepts in non-geometric setups.
9. Projections and Gram-Schmidt process in various inner product spaces
You will learn: apply Gram-Schmidt process in inner product spaces different from R^n (which were already covered in Part 2); work with projections on subspaces.
10. Min-max problems, best approximations, and least squares
You will learn: solve some simple min-max problems with help of Cauchy-Schwarz inequality, find the shortest distance to subspaces in IP spaces, handle inconsistent systems of linear equations.
C3 Symmetric matrices and quadratic forms
11. Diagonalization of symmetric matrices
You will learn: about various nice properties of symmetric matrices, and about orthogonal diagonalization.
12. Quadratic forms and their classification
You will learn: how to describe (geometrically) and recognise (from their equation) quadratic curves and surfaces.
13. Constrained optimization
You will learn: how to determine the range of quadratic forms on (generalized) unit spheres in R^n.
C4 The Grand Finale
14. Singular value decomposition
You will learn: about singular value decomposition: how it works and why it works; about pseudo-inverses.
15. Wrap-up Linear Algebra and Geometry
Also make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.
A detailed description of the content of the course, with all the 200 videos and their titles, and with the texts of all the 144 problems solved during this course, is presented in the resource file “List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_3.pdf” under video 1 ("Introduction to the course"). This content is also presented in video 1.