Linear Algebra and Geometry 3

Linear Algebra and Geometry 3
MP4 | Video: h264, 1280×720 | Audio: AAC, 44.1 KHz, 2 Ch

Inner product spaces, quadratic forms, and more advanced problem solvingWhat you’ll 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.
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.
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.
Quadratic forms and their connection to symmetric matrices: uniqueness of this correspondence and its consequences.
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.

Requirements
High-school and college mathematics (mainly arithmetics, some trigonometry, polynomials)
Linear Algebra and Geometry 1 (systems of equations, matrices and determinants, vectors and their products, analytic geometry of lines and planes)
Linear Algebra and Geometry 2 (vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors, diagonalization)
Some basic calculus
Basic knowledge of complex numbers (this course contains a short introduction to complex numbers)

Description
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

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.

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

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

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.

Who this course is for
University and college engineering

More info