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Course Contents

Matrices: matrix operations (Addition, Scalar Multiplication, Multiplication, Transpose, Ad joint and their properties; Special types of matrices (Null, Identity, Diagonal, Triangular, Symmetric, Skew Symmetric, Hermitian, Skew Hermitian, Orthogonal, Unitary, Normal), Solution of the matrix Equation Ax b; Row reduced Echelon form, Determinants and their properties, Vector Space Rn(R);Subspaces; Linear Dependence/Independence; Basis; Standard Basis of Rn; Dimension; Coordinates with respect to a basis; Complementary Sub spaces; Standard Inner product; Norm; Gram Schmidt Orthogonalization Process; Generalization to the vector space Cn(C), Linear Transformation from Rn toRm (motivation, X* AX); Image of a basis identifies the linear transformation; Range Space and Rank; Null Space and Nullity; Matrix Representation of a linear transformation; Structure of the solutions of the matrix equation Ax b; Linear Operators on Rn and their representation as square matrices; Similar Matrices and linear operators; Invertible linear operators; Inverse of a non singular matrix; Cramers method to solve the matrix equation Ax b; Eigen values and eigenvectors of a linear operator; Characteristic Equation; Bounds on eigen values; Diagonalizability of a linear operator; Properties of eigen values and eigen vectors of Hermitian, skew Hermitian, Unitary, and Normal matrices (including symmetric, skew symmetric, and orthogonal matrices), Implication of diagonalaizability of the matrix A + AT in the real quadratic form XTAX;