Linear Algebra and Geometry 3

• 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).
• Inner product spaces different from R^n: space of continuous functions, spaces of polynomials, spaces of matrices.
• Work with geometric concepts as length (norm), distance, angles, and orthogonality in non-geometric setups.
• Pythagorean Theorem, Cauchy-Schwarz inequality, and triangle inequality in various inner product spaces.
• Orthogonal and orthonormal bases, and Gram-Schmidt process in various inner product spaces.
• Min-max problems using Cauchy-Schwarz inequality, Best Approximation Theorem, least squares solutions.
• Symmetric matrices and their properties; orthogonal diagonalization: how it is done and how to understand it geometrically.
• Positive/negative definite matrices, indefinite matrices; various methods of determining definiteness of matrices.
• Quadratic forms and their connection to symmetric matrices: uniqueness of this correspondence and its consequences.
• Geometry of quadratic forms in two and three variables: conic sections and quadratic surfaces.
• Some concepts from abstract algebra: group, ring, field, and isomorphism; understand the concept of isomorphic vector spaces.
• Crowning of the course and a natural consequence of all the other topics: Singular Value Decomposition and pseudoinverses.
• Note: all the vector spaces discussed in this course are spaces over R (not over the field of complex numbers), and all our matrices have only real entries.

Linear Algebra and Geometry 3

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

Chapter 1: Eigendecomposition, spectral decomposition

S1. Introduction to the course

S2. 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.

S3. More problem solving; spaces different from R^n

You will learn: work with eigendecomposition of matrices for linear operators on various vector spaces.

S4. Intermezzo: isomorphic vector spaces

You will learn: about certain similarities between different spaces and how to measure them.

S5. Recurrence relations, dynamical systems, Markov matrices

You will learn: more exciting applications of eigenvalues and diagonalization.

S6. 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.

Chapter 2: Inner product spaces

S7. 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.

S8. Norm, distance, angles, and orthogonality in inner product spaces

You will learn: how to define geometric concepts in non-geometric setups.

S9. 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.

S10. 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.

Chapter 3: Symmetric matrices and quadratic forms

S11. Diagonalization of symmetric matrices

You will learn: about various nice properties of symmetric matrices, and about orthogonal diagonalization.

S12. Quadratic forms and their classification

You will learn: how to describe (geometrically) and recognise (from their equation) quadratic curves and surfaces.

S13. Constrained optimization

You will learn: how to determine the range of quadratic forms on (generalized) unit spheres in R^n.

Chapter 4: The Grand Finale

S14. Singular value decomposition

You will learn: about singular value decomposition: how it works and why it works; about pseudo-inverses.

S15. Wrap-up Linear Algebra and Geometry

S16. Extras

You will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.

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

“001 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.