Description

Linear Algebra and Geometry 2 – Complete Course Overview

Requirements

Before starting this course, students should be familiar with:

Foundational Mathematics

• Linear Algebra and Geometry 1
  (Systems of equations, matrices, determinants, vectors, analytic geometry of lines and planes)

• High-school and early college mathematics
  (Arithmetic, basic trigonometry, polynomial functions)

Additional Knowledge

• Basic Calculus
  (Used only in a few examples—can be skipped and revisited later)

  • Simple derivatives in: Videos 38, 39, 40, 132, 134, 136

  • Continuity mentioned in: Videos 16, 35–40

• Basic complex numbers
  (Used in an example in Video 8)

You are encouraged to ask questions anytime. If any concept in the lecture feels unclear, use the Q&A section so all students benefit from the explanations.

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Course Description: Linear Algebra and Geometry 2

This advanced course builds on the foundations of Linear Algebra and covers:

• Abstract vector spaces

• Matrix theory and transformations

• Orthogonality

• Eigenvalues, eigenvectors, and diagonalization

• Applied geometric and algebraic concepts used in engineering and mathematics

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Course Curriculum

Chapter 1: Abstract Vector Spaces

S1. Introduction to the Course

S2. Real Vector Spaces and Subspaces

You will learn:

• Definition of vector spaces and their axioms

• Determining subspaces

• Logical reasoning within vector space structures

S3. Linear Combinations & Linear Independence

You will learn:

• Linear combinations, span, and generating sets

• Linear independence vs. dependence

• Using Gaussian elimination to test independence

• Geometric intuition behind dependence/independence

S4. Coordinates, Basis & Dimension

You will learn:

• What a basis is and how coordinates work

• Understanding the dimension of vector spaces

• Using determinants to check if vectors form a basis in ℝⁿ

S5. Change of Basis

You will learn:

• Converting coordinates between different bases

• Transition matrices and their applications

• Solving systems of linear equations efficiently

• The geometry behind coordinate transformations

S6. Row Space, Column Space & Nullspace

You will learn:

• Row space, column space, and nullspace of a matrix

• Finding bases for vector spans in ℝⁿ

S7. Rank, Nullity & Fundamental Matrix Spaces

You will learn:

• Computing rank and nullity

• Orthogonal complements

• Understanding the four fundamental matrix subspaces

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Chapter 2: Linear Transformations

S8. Matrix Transformations from ℝⁿ to ℝᵐ

You will learn:

• How linear transformations correspond to matrices

• Kernel, image, and inverse transformations

• Relationship between transformations and matrix spaces

S9. Geometry of Matrix Transformations (ℝ² & ℝ³)

You will learn:

• Rotations, reflections, symmetries, projections

• Visualization of linear transformations

S10. Properties of Matrix Transformations

You will learn:

• Effects on subspaces, lines, and planes

• How transformations affect area and volume

• Matrix multiplication as transformation composition

S11. Transformations in Different Bases

You will learn:

• Solving problems involving transformations between vector spaces

• Working with operators using non-standard bases

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Chapter 3: Orthogonality

S12. Gram–Schmidt Process

You will learn:

• Orthonormal bases and their advantages

• Orthogonal projections in ℝⁿ

• Constructing orthonormal bases with Gram–Schmidt

S13. Orthogonal Matrices

You will learn:

• Properties and definitions of orthogonal matrices

• Their geometric interpretations

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Chapter 4: Introduction to Eigendecomposition

S14. Eigenvalues & Eigenvectors

You will learn:

• Computing eigenvalues and eigenvectors

• Geometric meaning of eigenvectors

• Understanding eigenspaces

S15. Diagonalization

You will learn:

• Checking matrix diagonalizability

• Diagonalizing matrices step-by-step

• Using diagonalization for fast matrix power calculations

S16. Course Wrap-Up

You will learn:

• An overview of topics covered

• Insight into the third course in this series

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Additional Course Resources

A complete index of all 214 videos, including detailed titles and all 153 solved problems, is available in:

“001_List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_2.pdf”
(Linked under Video 1: Introduction to the course)

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Who This Course Is For

• University and college engineering students

• Computer science and data science students

• Mathematics majors

• Anyone who wants a deeper understanding of Linear Algebra

• Learners preparing for advanced STEM coursework

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.