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# Applied Linear Algebra

## Syllabus

Applied Linear Algebra _ Course Introduction.
Vector Spaces: Introduction.
Linear Combinations and Span.
Subspaces, Linear Dependence and Independence.
Basis and Dimension.
Sums, Direct Sums and Gaussian Elimination.
Linear Maps and Matrices.
Null space, Range, Fundamental theorem of linear maps.
Column space, null space and rank of a matrix.
Algebraic operations on linear maps.
Invertible maps, Isomorphism, Operators.
Solving Linear Equations.
Elementary Row Operations.
Translates of a subspace, Quotient Spaces.
Row space and rank of a matrix.
Determinants.
Coordinates and linear maps under a change of basis.
Simplifying matrices of linear maps by choice of basis.
Polynomials and Roots.
Invariant subspaces, Eigenvalues, Eigenvectors.
More on Eigenvalues, Eigenvectors, Diagonalization.
Eigenvalues, Eigenvectors and Upper Triangularization.
Properties of Eigenvalues.
Linear state space equations and system stability.
Discrete-time Linear Systems and Discrete Fourier Transforms.
Sequences and counting paths in graphs.
PageRank Algorithm.
Dot product and length in Cn, Inner product and norm in V over F.
Orthonormal basis and Gram-Schmidt orthogonalisation.
Linear Functionals, Orthogonal Complements.
Orthogonal Projection.
Projection and distance from a subspace.
Linear equations, Least squares solutions and Linear regression.
Minimum Mean Squared Error Estimation.
Properties of Adjoint of a Linear Map.
Normal Operators.
Complex Spectral Theorem.
Real Spectral Theorem.
Positive Operators.
Quadratic Forms, Matrix Norms and Optimization.
Isometries.
Classification of Operators.
Singular Values and Vectors of a Linear Map.
Singular Value Decomposition.
Polar decomposition and some applications of SVD.

NPTEL-NOC IITM

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