## Advanced Matrix Theory and Linear Algebra for Engineers

Advanced Matrix Theory and Linear Algebra for Engineers. Instructor: Prof. Vittal Rao, Centre for Electronics Design and Technology, IISc Bangalore. Introduction to systems of linear equations, Vector spaces, Solutions of linear systems, Important subspaces associated with a matrix, Orthogonality, Eigenvalues and eigenvectors, Diagonalizable matrices, Hermitian and symmetric matrices, General matrices.
(from **nptel.ac.in**)

**Prologue** |

Lecture 01 - Prologue Part 1: Systems of Linear Equations, Matrix Notation |

Lecture 02 - Prologue Part 2: Diagonalization of a Square Matrix |

Lecture 03 - Prologue Part 3: Homogeneous Systems, Elementary Row Operations |

**Linear Systems** |

Lecture 04 - Linear Systems 1: Elementary Row Operations (EROs) |

Lecture 05 - Linear Systems 2: Row Reduced Echelon Form, The Reduction Process |

Lecture 06 - Linear Systems 3: The Reduction Process, Solution using EROs |

Lecture 07 - Linear Systems 4: Solution using EROs: Non-homogeneous Systems |

**Vector Spaces** |

Lecture 08 - Vector Spaces Part 1 |

Lecture 09 - Vector Spaces Part 2 |

**Linear Independence and Subspaces** |

Lecture 10 - Linear Combination, Linear Independence and Dependence |

Lecture 11 - Linear Independence and Dependence, Subspaces |

Lecture 12 - Subspace Spanned by a Finite Set of Vectors, The Basic Subspaces Associated with a Matrix |

Lecture 13 - Subspace Spanned by an Infinite Set of Vectors, Linear Independence of an Infinite Set of Vectors |

**Basis** |

Lecture 14 - Basis, Basis as a Maximal Linearly Independent Set |

Lecture 15 - Finite Dimensional Vector Spaces |

Lecture 16 - Extension of a Linearly Independent Set to a Basis, Ordered Basis |

**Linear Transformations** |

Lecture 17 - Relation between Representation in Two Bases, Linear Transformations |

Lecture 18 - Examples of Linear Transformations |

Lecture 19 - Null Space and Range of a Linear Transformation |

Lecture 20 - Rank Nullity Theorem, One-One Linear Transformation |

Lecture 21 - One-One Linear Transformation, Onto Linear Transformations, Isomorphisms |

**Inner Product and Orthogonality** |

Lecture 22 - Inner Product and Orthogonality |

Lecture 23 - Orthonormal Sets, Orthonormal Basis and Fourier Expansion |

Lecture 24 - Fourier Expansion, Gram-Schmidt Orthonormalization, Orthogonal Complements |

Lecture 25 - Orthogonal Complements, Decomposition of a Vector, Pythagoras Theorem |

Lecture 26 - Orthogonal Complements in the context of Subspaces Associated with a Matrix |

Lecture 27 - Best Approximation |

**Diagonalization** |

Lecture 28 - Diagonalization, Eigenvalues and Eigenvectors |

Lecture 29 - Eigenvalues and Eigenvectors, Characteristic Polynomial |

Lecture 30 - Algebraic Multiplicity, Eigenvectors, Eigenspaces and Geometric Multiplicity |

Lecture 31 - Criterion for Diagonalization |

**Hermitian and Symmetric Matrices** |

Lecture 32 - Hermitian and Symmetric Matrices, Unitary Matrix |

Lecture 33 - Unitary and Orthogonal Matrices, Eigen Properties of Hermitian Matrices, Unitary Diagonalization |

Lecture 34 - Spectral Decomposition |

Lecture 35 - Positive and Negative Definite and Semidefinite Matrices |

**Singular Value Decomposition (SVD)** |

Lecture 36 - Singular Value Decomposition (SVD) Part 1 |

Lecture 37 - Singular Value Decomposition (SVD) Part 2 |

**Back to Linear Systems** |

Lecture 38 - Back to Linear Systems Part 1 |

Lecture 39 - Back to Linear Systems Part 2 |

**Epilogue** |

Lecture 40 - Epilogue |