Linear Algebra

  1. Vector Spaces and Subspaces
    1. Vectors in Euclidean Space
      1. Definition of vectors in Rⁿ
        1. Vector notation and components
          1. Vector arithmetic
            1. Vector addition
              1. Scalar multiplication
              2. Geometric interpretation of vectors
                1. Linear combinations
                  1. Span of vector sets
                  2. Abstract Vector Spaces
                    1. Definition and axioms
                      1. Closure properties
                        1. Examples of vector spaces
                          1. Euclidean spaces Rⁿ
                            1. Matrix spaces
                              1. Polynomial spaces
                                1. Function spaces
                                2. Non-examples and common misconceptions
                                3. Subspaces
                                  1. Definition and subspace test
                                    1. Trivial subspaces
                                      1. Column space of a matrix
                                        1. Null space of a matrix
                                          1. Row space of a matrix
                                            1. Subspace operations
                                              1. Intersection of subspaces
                                                1. Sum of subspaces
                                              2. Linear Independence and Dependence
                                                1. Definition of linear independence
                                                  1. Definition of linear dependence
                                                    1. Testing for linear independence
                                                      1. Linear independence in Rⁿ
                                                        1. Maximal linearly independent sets
                                                        2. Basis and Dimension
                                                          1. Definition of basis
                                                            1. Standard basis vectors
                                                              1. Coordinate representation
                                                                1. Uniqueness of coordinate representation
                                                                  1. Dimension of vector spaces
                                                                    1. Dimension of subspaces
                                                                      1. Basis construction methods
                                                                        1. Extending linearly independent sets
                                                                          1. Reducing spanning sets
                                                                        2. Fundamental Subspaces
                                                                          1. Four fundamental subspaces
                                                                            1. Column space
                                                                              1. Null space
                                                                                1. Row space
                                                                                  1. Left null space
                                                                                  2. Relationships between subspaces
                                                                                    1. Rank-nullity theorem
                                                                                      1. Computing rank and nullity

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                                                                                    3. Determinants

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                                                                                    5. Linear Transformations