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