Linear Algebra and Geometry 3
Inner product spaces, quadratic forms, symmetric matrices: more advanced problem solving; singular value decomposition
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51 hours
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Jan 2025
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$69.99
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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).
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.
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4269888
udemy ID
8/29/2021
course created date
10/24/2021
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