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Linear Algebra: Matrices, Systems of Equations, and Eigenvectors

Explore the fundamental concepts of linear algebra, including matrices, systems of linear equations, determinants, vector spaces, orthogonal transformations, eigenvalues and eigenvectors. Understand their properties and applications in various fields.

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Linear Algebra: Matrices, Systems of Equations, and Eigenvectors

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  1. Introduction • The central problems of Linear Algebra are to study the properties of matrices and to investigate the solutions of systems of linear equations • Given A and b, find x for Ax=b; i.e., solve Ax=b • Given A, find λ (lambda) such that Ax=λ x

  2. Matrices and Linear Systems of Equations • Matrix notations and operations • Elementary row operations • Row-echelon form • Matrix inverse • LU-Decomposition for Invertible Matrices • LU-Decomposition with Partial Pivoting • Some special matrices

  3. Determinants • Definition of Determinant det(A) or |A| • Cofactor and minor at (i,j)-position of A • Properties of determinants • Examples and Applications

  4. Vector Space and Linear Transform • Vector Space, Subspace, Examples • Null space, column space, row space of a matrix • Spanning sets, Linear Independence, Basis, Dimension • Rank, Nullity, and the Fundamental Matrix Spaces • Matrix Transformation from Rn to Rm • Kernel and Image of a Linear Transform Projection, rotation, scaling Gauss transform Householder transform (elementary reflector) Jacobi transform (Givens’ rotation) • An affine transform is in general not a linear transform Vector norms and matrix norms

  5. Orthogonality • Motivation – More intuitive • Inner product and projection • Orthogonal vectors and linear independence • Orthogonal complement • Projection and least squares approximation • Orthonormal bases and orthogonal matrices • Gram-Schmidt orthogonalization process • QR factorization

  6. Eigenvaluesand Eigenvectors • Definitions and Examples • Properties of eigenvalues and eigenvectors • Diagonalization of matrices • Similarity transformation and triangularization Positive definitive (Semi-definite) matrices • Spectrum decomposition • Singular Value Decomposition (SVD) • A Markov process • Differential equations and exp(A) • Algebraic multiplicity and geometric multiplicity • Minimal polynomials and Jordan blocks

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