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The exhaustive list of topics in Matrix Theory And Linear Algebra in which we provide Help with Homework Assignment and Help with Project is as follows:

  • First Basic Problem – Systems of Linear equations - Matrix Notation – The various questions that arise with a system of linear eqautions
  • Second Basic Problem – Diagonalization of a square matrix – The various questions that arise with diagonalization.
  • Vector Spaces
  • Vector spaces
  • Subspaces
  • Linear combinations and subspaces spanned by a set of vectors
  • Linear dependence and Linear independence
  • Spanning Set and Basis
  • Finite dimensional spaces
  • Dimension
  • Solutions of Linear Systems
  • Simple systems
  • Homogeneous and Nonhomogeneous systems
  • Gaussian elimination
  • Null Space and Range
  • Rank and nullity
  • Consistency conditions in terms of rank
  • General Solution of a linear system
  • Elementary Row and Column operations
  • Row Reduced Form
  • Triangular Matrix Factorization
  • Important Subspaces associsted with a matrix
  • Range and Null space
  • Rank and Nullity
  • Rank Nullity theorem
  • Four Fundamental subspaces
  • Orientation of the four subspaces
  • Orthogonality
  • Inner product
  • Inner product Spaces
  • Cauchy – Schwarz inequality
  • Norm
  • Orthogonality
  • Gram – Schmidt orthonormalization
  • Orthonormal basis
  • Expansion in terms of orthonormal basis – Fourier series
  • Orthogonal complement
  • Decomposition of a vector with respect to a subspace and its orthogonal complement – Pythagorus Theorem
  • Eigenvalues and Eigenvectors
  • Eigenvalue – Eigenvector pairs
  • Where do we look for eigenvalues – characteristic equation
  • Algebraic multiplicity
  • Eigenvectors, Eigenspaces and geometric multiplicity
  • Diagonalizable Matrices
  • Diagonalization criterion
  • Diagonalizing matrix
  • Cayley-Hamilton theorem, Annihilating polynomials, Minimal Polynomial
  • Diagonalizability and Minimal polynomial
  • Projections
  • Decomposition of the matrix in terms of projections
  • Hermitian Matrices
  • Real symmetric and Hermitian Matrices
  • Properties of eigenvalues and eigenvectors
  • Unitary/Orthoginal Diagonalizbility of Complex Hermitian/Real Symmetric matrices
  • Spectral Theorem
  • Positive and Negative Definite and Semi definite matrices
  • General Matrices
  • Matrices AAT and ATA
  • Rank, Nullity, Range and Null Space of AAT and ATA
  • Strategy for choosing the basis for the four fundamental subspaces
  • Singular Values
  • Singular Value Decomposition
  • Pseudoinverse and Optimal solution of a linear system of equations
  • Geometry of Pseudoinverse
  • Jordan Cnonical form
  • Primary Decomposition Theorem
  • Nilpotent matrices
  • Canonical form for a nilpotent matrix
  • Jordan Canonical Form
  • Functions of a matrix
  • Selected Topics in Applications
  • Optimization and Linear Programming
  • Network models
  • Game Theory
  • Control Theory
  • Image Compression