Description

Linear Algebra and Geometry 3 – Advanced Course Overview

Requirements

To get the most out of this advanced Linear Algebra course, you should already have foundational knowledge in:

Essential Mathematics Background

• High-school and early college mathematics
  (arithmetic, basic trigonometry, polynomials)

• Linear Algebra & Geometry 1:
  Systems of equations, matrices and determinants, vector operations, analytic geometry of lines and planes

• Linear Algebra & Geometry 2:
  Vector spaces, linear transformations, orthogonality, eigenvalues/eigenvectors, diagonalization

Optional but Helpful

• Some Calculus:
  Only required for specific examples involving derivatives or integrals (e.g., Sections 6–9 and Videos 18–19). These examples can be skipped and revisited later.

• Basic understanding of complex numbers
  (A brief introduction is included in the course.)

Student Support

If you have questions at any point, you are encouraged to ask. Using the course Q&A helps all learners benefit from shared explanations and clarifications.

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Course Description: Linear Algebra and Geometry 3

This advanced course explores inner product spaces, quadratic forms, eigendecomposition, spectral theory, and advanced problem solving—ideal for students aiming to deepen their understanding of Linear Algebra and its real-world applications.

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📘 Chapter 1: Eigendecomposition & Spectral Decomposition

S1. Introduction to the Course

S2. Geometrical Operators in 2D & 3D

You will learn:

• Using eigenvalues/eigenvectors to analyze symmetries, projections, rotations

• How to derive standard matrices of geometric operators

• Deeper insights into geometric transformations

S3. Problem Solving with Vector Spaces Beyond ℝⁿ

You will learn:

• Working with eigendecomposition in various abstract vector spaces

S4. Isomorphic Vector Spaces

You will learn:

• Understanding similarities between vector spaces

• Measuring and interpreting structural equivalence

S5. Recurrence Relations, Dynamical Systems, Markov Matrices

You will learn:

• Real-world applications of eigenvalues and diagonalization

S6. Solving Linear ODE Systems

You will learn:

• Solving systems of linear differential equations

• Handling higher-order ODEs using eigenvalues and diagonalization

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📘 Chapter 2: Inner Product Spaces

S7. Inner Products Beyond the Dot Product

You will learn:

• Alternative inner product definitions in different vector spaces

S8. Norms, Distances, Angles & Orthogonality

You will learn:

• Geometric concepts defined in non-geometric contexts

S9. Projections & Gram-Schmidt in General Inner Product Spaces

You will learn:

• Gram-Schmidt orthonormalization beyond ℝⁿ

• Working with projections on various subspaces

S10. Min-Max Problems & Least Squares

You will learn:

• Applying Cauchy-Schwarz inequality

• Best approximations in IP spaces

• Solving inconsistent systems using least squares

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📘 Chapter 3: Symmetric Matrices & Quadratic Forms

S11. Diagonalizing Symmetric Matrices

You will learn:

• Properties of symmetric matrices

• Orthogonal diagonalization techniques

S12. Quadratic Forms & Classification

You will learn:

• Recognizing and describing quadratic surfaces & curves

• Interpreting equations geometrically

S13. Constrained Optimization

You will learn:

• Finding ranges of quadratic forms on generalized unit spheres

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📘 Chapter 4: The Grand Finale

S14. Singular Value Decomposition (SVD)

You will learn:

• How SVD works and why

• Using SVD to compute pseudo-inverses

S15. Course Wrap-Up

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Additional Resources

A detailed list of:

• All 200 video lectures

• All 144 solved problems

• Titles and descriptions

is available in the PDF resource:
“001 List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_3.pdf” under Video 1 (“Introduction to the course”).

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Who This Course Is For

• University and college engineering students

• Learners pursuing mathematics, computer science, data science, physics, and related fields

• Anyone seeking a deeper understanding of advanced linear algebra concepts

Please Note: Files will be included in this purchase only Full Course Video & Course Resources. You will get cloud storage download link with life time download access.