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Notes on Mathematics -
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Contents
Contents
Linear Algebra
Subsections
Matrices
Definition of a Matrix
Special Matrices
Operations on Matrices
Multiplication of Matrices
Inverse of a Matrix
Some More Special Matrices
Submatrix of a Matrix
Block Matrices
Miscellaneous Exercises
Matrices over Complex Numbers
Linear System of Equations
Introduction
A Solution Method
Row Operations and Equivalent Systems
Gauss Elimination Method
Row Reduced Echelon Form of a Matrix
Gauss-Jordan Elimination
Elementary Matrices
Rank of a Matrix
Existence of Solution of
Example
Main Theorem
Equivalent conditions for Invertibility
Inverse and the Gauss-Jordan Method
Determinant
Adjoint of a Matrix
Cramer's Rule
Miscellaneous Exercises
Finite Dimensional Vector Spaces
Vector Spaces
Definition
Examples
Subspaces
Linear Combinations
Linear Independence
Bases
Important Results
Ordered Bases
Linear Transformations
Definitions and Basic Properties
Matrix of a linear transformation
Rank-Nullity Theorem
Similarity of Matrices
Inner Product Spaces
Definition and Basic Properties
Gram-Schmidt Orthogonalisation Process
Orthogonal Projections and Applications
Matrix of the Orthogonal Projection
Eigenvalues, Eigenvectors and Diagonalisation
Introduction and Definitions
Diagonalisation
Diagonalisable matrices
Sylvester's Law of Inertia and Applications
A K Lal 2007-09-12
