Linear Algebra

  1. Matrix Theory and Operations
    1. Introduction to Matrices
      1. Definition and notation
        1. Matrix dimensions and terminology
          1. Entry notation and indexing
            1. Row vectors and column vectors
              1. Special positions and pivot elements
                1. Matrix equality
                2. Basic Matrix Operations
                  1. Matrix addition
                    1. Definition and requirements
                      1. Properties of matrix addition
                      2. Matrix subtraction
                        1. Scalar multiplication
                          1. Definition and properties
                            1. Distributive properties
                            2. Matrix multiplication
                              1. Definition and row-column rule
                                1. Conditions for multiplication
                                  1. Properties of matrix multiplication
                                    1. Non-commutativity of multiplication
                                    2. Matrix transpose
                                      1. Definition and notation
                                        1. Properties of transpose operations
                                      2. Special Types of Matrices
                                        1. Zero matrix
                                          1. Identity matrix
                                            1. Diagonal matrices
                                              1. Upper triangular matrices
                                                1. Lower triangular matrices
                                                  1. Symmetric matrices
                                                    1. Skew-symmetric matrices
                                                      1. Block matrices and partitioned matrices
                                                      2. Matrix Inverses
                                                        1. Definition of matrix inverse
                                                          1. Properties of invertible matrices
                                                            1. Uniqueness of inverses
                                                              1. Computing inverses using Gauss-Jordan elimination
                                                                1. Inverses of 2×2 matrices
                                                                  1. Properties of inverse operations
                                                                    1. Singular and non-singular matrices
                                                                    2. Elementary Matrices and Factorizations
                                                                      1. Elementary matrices
                                                                        1. Row interchange matrices
                                                                          1. Row scaling matrices
                                                                            1. Row addition matrices
                                                                            2. Elementary matrices and row operations
                                                                              1. Inverses of elementary matrices
                                                                                1. LU factorization
                                                                                  1. Lower triangular matrices
                                                                                    1. Upper triangular matrices
                                                                                      1. Existence and computation of LU factorization
                                                                                        1. Applications of LU factorization

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                                                                                    1. Foundations of Linear Systems

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                                                                                    3. Determinants