For details of standard term assessment timelines and exam structures, visit our Academics page.
WEEK 1
Introduction, Real and complex numbers, Vectors, linear equations and linear combination in R^2, Span of a set of vectors in R^2, Vector spaces and Subspaces, Linear dependent and Independent vectors, Basis and dimension of vector (sub)space in R^2, Change of basis in R^2.
WEEK 2
Linear Transformations in R^2, Linear Transformation and Basis, Matrix representation of a Linear Transformation,Range, Null, and Rank of a Linear Transformation, Composition of Linear Transformation and Matrix Multiplication,Inverse of a linear transformation, Change of basis.
WEEK 3
Range and Null space of Composition of Linear transformation in R^2, Linear Operators, Applications of operators, Scaling and rotation, Basic matrix algebra and complex vector space, Eigenvalues, Eigenvectors, Properties of eigenvalues and eigenvectors, Diagonalization of a Matrix, Power of a Matrix.
WEEK 4
Inner product in R^2 and C^2, Magnitude of a vector, Distance between two vectors, and Angle between vectors, Orthogonal complement or Dual, Inner product as a Linear transformation, Adjoint of an operator, Range and Null of an Operator and its Adjoint, Operations with adjoint, Eigenvalues and Eigenvectors of an operator and its adjoint, Least square fit: Orthogonality principle.
WEEK 5
Vector spaces in R^n and C^n, Span of vectors, Linear dependence and independence, spanning sets and basis, Linear Transformations (in Higher dimensions), Range and Null of a Linear Transformation, Fundamental theorem of Linear Transformation
WEEK 6
Matrix Algebra - Addition, Multiplication, Inverse, and Powers, Eigenvalues and eigenvectors, Existence of eigenvalues, Determinants, Algebraic and Geometric multiplicities, Diagonalizability and its applications, Change of Basis, Upper Triangularization, Circular Convolution and Circular Matrix.
WEEK 7
Inner Product Spaces, Norm of vector, Orthogonal vectors, Orthogonal decomposition, Orthonormal basis and orthonormalization, Gram-schmidt orthonormalization, Orthonormal basis: Properties, Least Square Solutions,Orthogonal Complement and Projections.
WEEK 8
Operators in Inner product spaces, Adjoint of a linear transformation, Four fundamental spaces, Operators and their adjoints, Eigenvalues and Eigenvectors of operators, Self adjoint operators, Diagonalizability of self adjoint operators
WEEK 9
Conics in 2D, Quadratic forms and Symmetric matrices, Oscillation modes in system, Correlation and Covariance Matrix, Positive semi-definite matrices, Graph and Adjacency Matrix
WEEK 10
Circulant Matrices and Fourier Basis, Discrete Fourier Transform (DFT), Normal Operators and Matrices.
WEEK 11
Sequence and Function spaces, Inner product and Norm in Sequence and Function spaces, Orthonormal basis for Signal spaces, Polynomial Approximations and Projections in Function Spaces, Sinusoids, Projection, Fourier Series, Complex Exponential form of Fourier Series
