"great material is presented, a lot of exercises. Wish for better navigation and for the system status bar to be visible. It would be nice to have a version of the app for Windows."
152
Sub Topics
200
MCQs
86
MCOs
143
True/False
81
Fill Blanks
32
Rearrange
81
Matching
43
Comprehensions
81
Flashcard Decks
Curriculum
What You'll Learn
01 Introduction to Linear Algebra 2 topics
1 Fundamentals and Notation
- Mathematical notation and symbols
- Sets, fields, and vector spaces
- Brief history of linear algebra
2 Applications of Linear Algebra
- Engineering applications
- Computer science applications
- Physics and scientific applications
- Data science and machine learning applications
02 Vectors and Vector Spaces 4 topics
1 Vector Basics
- Definition of vectors
- Vector operations (addition, scalar multiplication)
- Geometric interpretation of vectors
2 Vector Spaces
- Definition and axioms
- Subspaces
- Span and linear independence
3 Vector Norms
- Definition of norms
- Common vector norms (L1, L2, Lā)
- Properties of norms
4 Inner Product Spaces
- Definition of inner product
- Properties of inner products
- Cauchy-Schwarz inequality
- Orthogonality and orthogonal projections
03 Matrices and Matrix Operations 3 topics
1 Matrix Basics
- Definition and notation
- Types of matrices (square, rectangular, diagonal, triangular)
- Special matrices (identity, zero, symmetric, skew-symmetric)
2 Matrix Operations
- Matrix addition and scalar multiplication
- Matrix multiplication
- Matrix transpose
- Trace of a matrix
3 Block Matrices
- Definition and notation
- Operations with block matrices
- Applications of block matrices
04 Systems of Linear Equations 4 topics
1 Introduction to Linear Systems
- Coefficient matrix and augmented matrix
- Consistent and inconsistent systems
- Homogeneous and non-homogeneous systems
2 Solution Methods
- Gaussian elimination
- Gauss-Jordan elimination
- LU decomposition
3 Matrix Representation
- Matrix form of linear systems (Ax = b)
- Solution spaces and general solutions
4 Computational Considerations
- Numerical stability
- Pivoting strategies
- Computational complexity
05 Linear Transformations 3 topics
1 Basics of Linear Transformations
- Definition and properties
- Kernel and range
- Matrix representation of linear transformations
2 Common Linear Transformations
- Rotation and reflection
- Scaling and shearing
- Projection transformations
3 Change of Basis
- Basis and coordinates
- Transition matrices
- Similar matrices
06 Determinants 3 topics
1 Definition and Properties
- Definition of determinants
- Determinant of special matrices
- Properties of determinants
2 Calculation Methods
- Cofactor expansion
- Row reduction method
- Properties-based calculation
3 Applications of Determinants
- Area and volume calculation
- Cramer's rule
- Testing for invertibility
07 Eigenvalues and Eigenvectors 3 topics
1 Fundamentals
- Definition of eigenvalues and eigenvectors
- Characteristic polynomial
- Algebraic and geometric multiplicity
2 Eigendecomposition
- Diagonalization
- Conditions for diagonalizability
- Application of diagonalization
3 Special Cases
- Symmetric matrices and orthogonal eigenvectors
- Positive definite matrices
- Defective matrices
08 Vector Spaces with Inner Products 3 topics
1 Inner Product Spaces
- Definition and properties
- Examples of inner products
- Gram-Schmidt orthogonalization process
2 Orthogonality
- Orthogonal and orthonormal bases
- Orthogonal complements
- Orthogonal projections
3 Least Squares
- Best approximation theorem
- Normal equations
- Applications in data fitting
09 Singular Value Decomposition (SVD) 3 topics
1 SVD Theory
- Definition and existence theorem
- Geometric interpretation
- Properties of singular values
2 Computation of SVD
- Algorithms for SVD computation
- Truncated SVD
- Numerical considerations
3 Applications of SVD
- Image compression
- Data analysis and dimensionality reduction
- Signal processing
10 Jordan Canonical Form 3 topics
1 Jordan Blocks and Jordan Form
- Definition of Jordan blocks
- Construction of Jordan canonical form
- Relationship with eigenvalues and eigenvectors
2 Computing Jordan Form
- Generalized eigenvectors
- Algorithm for finding Jordan form
- Examples and special cases
3 Applications
- Solving systems of differential equations
- Matrix powers and exponentials
- Analysis of dynamic systems
11 Quadratic Forms and Definiteness 3 topics
1 Quadratic Forms
- Definition and matrix representation
- Classification of quadratic forms
- Change of variables
2 Definiteness
- Positive and negative definiteness
- Semi-definiteness
- Indefinite forms
3 Applications
- Optimization problems
- Stability analysis
- Statistical applications
12 Linear Algebra in Function Spaces 3 topics
1 Function Spaces as Vector Spaces
- Continuous and differentiable function spaces
- Inner products of functions
- Orthogonal function systems
2 Fourier Series and Transforms
- Fourier series expansion
- Connection to linear algebra
- Discrete and fast Fourier transforms
3 Wavelets and Function Approximation
- Wavelet basis functions
- Multi-resolution analysis
- Applications in signal processing
13 Computational Linear Algebra 3 topics
1 Numerical Methods
- Iterative methods for large systems
- Krylov subspace methods
- Preconditioning techniques
2 Error Analysis
- Conditioning of problems
- Stability of algorithms
- Floating-point considerations
3 Software and Implementation
- Linear algebra libraries
- Parallel algorithms
- GPU acceleration
14 Applications of Linear Algebra 4 topics
1 Computer Graphics
- 3D transformations
- Rendering pipelines
- Animation and modeling
2 Machine Learning and Data Analysis
- Principal Component Analysis
- Linear regression
- Neural networks and linear layers
3 Control Theory
- State-space representation
- Controllability and observability
- Stability analysis
4 Quantum Mechanics
- Hilbert spaces
- Hermitian operators
- Quantum computing basics
15 Advanced Topics 3 topics
1 Tensor Algebra
- Introduction to tensors
- Tensor operations
- Applications in physics and engineering
2 Sparse Matrices
- Storage formats
- Specialized algorithms
- Applications with large-scale data
3 Randomized Linear Algebra
- Random projections
- Sketching techniques
- Probabilistic algorithms
16 Appendices 3 topics
1 Mathematical Foundations
- Set theory review
- Real and complex number properties
- Logic and proof techniques
2 Algorithms and Pseudocode
- Matrix operations implementation
- Decomposition algorithms
- Optimization routines
3 Reference Tables
- Common matrix properties
- Algorithm complexity summary
- Special matrices and their properties
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Linear Algebra
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