Echelon Form of a Matrix

Unit 1: Introduction to Matrix Algebra

• Echelon Form of a Matrix
• The Rank of a Matrix
• Determinant and Inverse of a Matrix
• Solution of System of Homogeneous & Non- Homogeneous Equations: Gauss Elimination
• Solution of System of Homogeneous Equations: Gauss Jordan Method

Unit 2: Vector Space and Linear Transformation

• Vector Space
• Sub-spaces
• Linear Combinations
• Linear Span
• Convex Sets
• Linear Independence Dependence
• Basis & Dimension
• Linear Transformation on Finite Dimensional
• Vector Spaces
• Inner Product Space
• Schwarz Inequality
• Orthonormal Basis
• Gram-schmidt Orthogonalization Process

Unit 3: Eigenvalue and Eigenvector

• Characteristic Polynomial
• Cayley Hamilton Theorem
• Eigen Value and Eigen Vector of a Matrix
• Eigenspaces
• Diagonalization
• Positive Definite Matrices
• Applications to Markov Matrices

Unit 4: Vector Calculus

• Vector Algebra
• Laws of Vector Algebra
• Dot Product
• Cross Product
• Vector and Scalar Fields
• Ordinary Derivative of Vectors
• Space Curves
• Partial Derivatives
• Del Operator
• Gradient of a Scalar Field
• Directional Derivative
• Gradient of Matrices
• Divergence of a Vector Field
• Laplacian Operator
• Curl of a Vector Field