Linear algebra is the study of lines, planes, and subspaces, but it's also the language of many scientific fields
MATRICES AND SYSTEM OF LINEAR EQUATIONS [14 Hours] Row-reduced echelon matrices – Rank of matrix– Normal form – Constructing non-singular Matrices – Sparse matrices. Introduction to System of linear equations – representation in matrix form – consistency – solving homogeneous and non- homogeneous system of linear equations. Gauss elimination and Gauss-Jordan method
MATRIX DECOMPOSITIONS: [10 Hours] LU decomposition – Cholesky decomposition – Eigenvalues - Eigenvectors for symmetric, skew-symmetric and orthogonal Matrices – properties of eigenvalues and eigen vectors (without proofs) – Diagonalization -singular value decomposition
CAYLEY-HAMILTON THEOREM AND REAL QUADRATIC FORM: [8 Hours] Cayley-Hamilton theorem – Inverse and power of matrix by using Cayley-Hamilton theorem – Real Quadratic form – Linear transformation of a Q.F.- Canonical form of a real Q.F – Reduction of Quadratic form to canonical form by using orthogonal reduction – Rank, index, signature and Nature of quadratic form.
VECTOR SPACES AND INNER PRODUCT SPACES [8 Hours] Vector spaces – subspaces – Linear span – linear dependency –– linear Transformations as Vectors and Matrices – Inner product spaces – Norm of vector – Orthogonal and Orthonormal sets – Gram-Schmidt Orthogonalization process.
VECTORS AND TENSORS [8 Hours] Scalar point function – vector point function – Vector differential operator – Gradient – directional derivative, angle between two surfaces- Divergence - Solenoidal vector - Curl - Irrotational Field - scalar potential.Tensor - Covariant and Contravariant Tensors.