Linear Algebra
- •Introduction to vectors (vector addition, scalar multiplication, linear combination of vectors, inner product, vector norm, distance, angle, orthogonality)
- •Introduction to Matrices (matrix addition and multiplication, matrix derivative and integral, matrix transpose, trace, identity matrix, block matrix , matrix polynomial, determinant, minor and cofactor, singular matrices, adjoint matrix, inverse matrix
- •Introduction to Matrices 2( symmetric/ skew-symmetric/ orthogonal/ conjugate/ hermitian/ skew-hermitian/ normal/ unitary matrices, matrix norm, positive/negative (semi-)definite matrices, quadratic form, Silvester criterion)
- •Linear equations1 (Homogeneous system, Augmented matrix, inconsistency, ill-condition, condition number, Gaussian elimination, Backward substitution algorithm, Elementary matrix, Elementary matrix ,
- •Linear equations2 ( Gauss Jordan, Row echelon form, pivoting, reduced row echelon form, LU factorization, LU factorization with pivoting, Cholesky factorization) , Applications of linear systems in network analysis, electrical circuits and chemical equations, polynomial interpolation, economics (Leontief Input-Output Models)
- •Vector spaces ( Field & vector space, subspace, Linear combination, Column/Row space, Spanning sets, linear independency, Basis, Dimension, Rank, Coordinate, Range space, Null space, Nullity, Fundamental subspaces)
- •Linear transformation ( functions, one-to-one function, surjective (onto) function, function composition, Matrix functions, identity and inverse transformation, Linear transformation, matrix linear transformations, linear transformation null space and range space, rank and nullity of a linear transformation, Isomorphism, Similar matrices, Similarity transformation)
- •Eigenvalues and eigenvectors (Eigenvalues, Eigenvectors, Characteristic equation, Monic, Cayley-Hamilton theorem , Power method, QR factorization, Diagonal form, Block diagonal form, Companion form, Jordan canonical form, minimal polynomial, Applications (Markov chain, Differential equations)
- Inner product vector spaces, orthogonality, Least-square problem (inner product spaces, Orthogonal complement, Orthogonal basis, Orthonormal basis, Gram-Schmidth process, Orthogonal projection, Least- square problem, Normal equations, QR factorization, Cholesky factorization), mathematical modelling using least squares, Function approximation (fourier series)
- •Singular value decomposition (Singular values, Singular value decomposition (SVD), left and right singular vectors, rank/ 2-norm/ determinant/ inverse matrix computation based on SVD, Pseudo-inverse), ,Applications (Data compression, PCA algorithm)
- •Matrix polynomials & functions (Matrix polynomials, Matrix functions, inverse matrix computation, State-space representation, Similar realizations, State transition matrix, methods for computation of state transition matrix)