Linear Algebra and Geometry 3

Linear Algebra and Geometry 3

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.

Inner product spaces, quadratic forms, and more advanced problem solving

Url: View Details

What you will learn

• How to solve problems in linear algebra and geometry (illustrated with 144 solved problems) and why these methods work.
• Solve more advanced problems on eigendecomposition and orthogonality than in the second course.
• Use diagonalization of matrices for solving various problems from different branches of mathematics (ODE, dynamical systems).

Rating: 4.95455

Level: Intermediate Level

Duration: 51 hours

Instructor: Hania Uscka-Wehlou