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Home/Linear Algebra

Linear Algebra

Master the language of modern mathematics and science. From abstract vector spaces to practical matrix computations, build a deep understanding of linear structures.

Our Approach

This course follows the modern "Linear Maps before Matrices" pedagogy, emphasizing conceptual understanding over rote computation. We start with abstract vector spaces, develop the theory of linear transformations, and only then introduce matrices as representations of these maps.

Abstract thinking: understand the essence, not just the computation
Rigorous proofs: build mathematical maturity
Real applications: connect theory to practice

Course Chapters

Part 1

Foundations

Available

Essential prerequisites: algebraic structures (groups, fields), complex numbers, equivalence relations, and Gaussian elimination

Foundation10-12 hours4 Courses

Topics Covered

Algebraic Structures

Complex Numbers

Equivalence Relations

Gaussian Elimination

Part 2

Vector Spaces

Available

Abstract vector spaces over fields, subspaces, linear independence, basis, dimension, and direct sums

Core14-18 hours5 Courses

Topics Covered

Vector Space Definition

Subspaces

Linear Independence

Basis & Dimension

Direct Sums

Part 3

Linear Mappings

Available

Linear transformations, kernel and image, rank-nullity theorem, isomorphisms, and dual spaces

Core16-20 hours5 Courses

Topics Covered

Linear Map Definition

Kernel & Image

Rank-Nullity Theorem

Isomorphisms

Dual Spaces

Part 4

Matrix Theory

Available

Matrix representation of linear maps, matrix operations, inverses, elementary matrices, and rank

Core18-22 hours6 Courses

Topics Covered

Matrix Representation

Matrix Operations

Matrix Inverse

Elementary Matrices

Rank & Equivalence

Part 5

Determinants

Available

Axiomatic and computational approaches to determinants, Laplace expansion, and adjugate matrices

Core12-15 hours5 Courses

Topics Covered

Determinant Definition

Properties

Computation Methods

Laplace Expansion

Adjugate Matrix

Part 6

Eigenvalues & Eigenvectors

Available

Eigenvalue theory, characteristic polynomials, diagonalization, Jordan normal form, and Cayley-Hamilton theorem

Advanced20-25 hours5 Courses

Topics Covered

Eigenvalue Definition

Characteristic Polynomial

Diagonalization

Jordan Form

Cayley-Hamilton

Part 7

Inner Product Spaces

Available

Inner products, orthogonality, Gram-Schmidt process, orthogonal projections, and spectral theorem

Advanced18-22 hours5 Courses

Topics Covered

Inner Product Definition

Orthogonality

Gram-Schmidt Process

Projections

Spectral Theorem

Part 8

Advanced Topics

Available

Singular value decomposition, quadratic forms, tensors, and real-world applications

Advanced16-20 hours4 Courses

Topics Covered

Singular Value Decomposition

Quadratic Forms

Tensors & Multilinear Algebra

Applications

Learning Path

Foundations

4 courses

Vector

5 courses

Linear

5 courses

Matrix

6 courses

Determinants

5 courses

Eigenvalues

5 courses

Inner

5 courses

Advanced

4 courses

Why Study Linear Algebra?

🤖 Machine Learning & AI

Neural networks, dimensionality reduction, and recommendation systems all rely on linear algebra.

🎮 Computer Graphics

3D transformations, rotations, and projections are all matrix operations.

⚛️ Quantum Computing

Quantum states are vectors, and quantum gates are unitary matrices.

📊 Data Science

PCA, SVD, and linear regression are fundamental tools for data analysis.

🔐 Cryptography

Many encryption schemes use matrix operations over finite fields.

📐 Pure Mathematics

The gateway to abstract algebra, functional analysis, and representation theory.

Ready to Master Linear Algebra?

Begin your journey with the foundations and discover the elegant structure underlying modern mathematics

Course content coming soon!