Linear Algebra

Table of Contents

1. System of Linear Equations
   • System of Linear Equations
   • Augmented Matrix
   • Row Operation
     • Gaussian elimination and back substitution Extra: The three operations do not change solutions
     • Row operation
   • Row Echelon Form
     • Row echelon form
     • Shape of row echelon form
     • Computing row echelon form
     • Reduced row echelon form Extra: Reduced row echelon form vs. General solution
   • Existence
     • Existence of solutions for fixed right side
     • Existence of solutions for any right side
   • Uniqueness
     • Structure of solutions
     • Uniqueness of solutions
   • A Basic Linear Algebra Principle
     • Implication of existence/uniqueness on size
     • Implication of size on existence/uniqueness
2. Linear Transformation and Matrix
   • Vector and Matrix
     • Euclidean vector and euclidean space Extra: Vector addition is given by parallelogram
     • Matrix
     • Some matrix notations and terminologies
   • Linear Transformation
     • Transformation Extra: Root-Polynomial transformation
     • Matrix transformation
     • Linear transformation Extra: Matrix transformations are linear
     • Linearity in picture
     • Linear transformation = matrix transformation Extra: Matrix of a linear transformation
   • Onto
     • Range and existence
     • Onto and always existence
   • One-to-one
     • One-to-one and uniqueness
     • Kernel and uniquess
   • Composition and Matrix Multiplication
     • Composition of transformations
     • Multiplication of matrices Extra: Composition of linear transformations is linear
     • Properties of matrix multiplication
     • Composition/multiplication mixed with addition and scalar multiplication
   • Inverse
     • Inverse transformation Extra: Criterion for invertibility of a transformation
     • Inverse matrix
     • Inverse of 2 by 2 matrix
     • Computation of inverse matrix
     • Properties of inverse Extra: Criterion for invertibility of a matrix
   • Transpose
   • Partitioned Matrix
   • Appendix: Transformation
     • Definition of transformation
     • Onto
     • One-to-one
     • Inverse
     • Graph of transformation
     • Formal theory of transformation
3. Vector Space
   • Vector Space
     • Motivation
     • How to make the definition
     • Definition
     • Proofs in vector space
   • Subspace
     • Definition
     • Sum and intersection
   • Linear Transformation
     • Definition
     • Operations of linear transformations and Hom space
   • Range and Column Space
     • Range
     • Column space
   • Kernel and Null Space
     • Kernel
     • Null space
     • Structure of solutions of general linear equations
   • Span
     • Definition and relation to existence
     • Geometric intuition of span
     • Properties of span Extra: Properties of span
   • Linear Independence
     • Definition and relation to uniqueness
     • Geometric intuition of linear independence
     • Properties of independence Extra: Properties of linear independence
   • Basis
     • Definition
     • Coordinate
     • Isomorphism
     • Dimension
     • Properties of dimension
   • Computation of Basis
     • Basis of null space
     • Basis of column space, first method
     • Basis of column space, second method
   • Rank and Nullity
     • Rank and nullity for matrix
     • Properties of rank and nullity Extra Properties of rank and nullity
     • Rank and nullity for linear transformation
   • Change of Basis
     • Matrix for change of coordinates
     • Computing the matrix for change of coordinates
     • Matrix for linear transformation
     • Change of the matrix for linear transformation
     • Similar matrix
   • Dual
     • Dual space
     • Dual basis
     • Dual transformation
     • Double dual
   • Direct Sum
     • Direct sum of vector spaces
     • Direct sum of subspaces
     • Properties of direct sum
     • Direct summand
     • Linear transformation between direct sums
   • Appendix: Field
     • Number system and field
     • Complex number
     • Linear algebra over a field*
4. Determinant
   • Geometry of Determinant
     • Determinant of 2 by 2 matrix
     • Determinant of 3 by 3 matrix
     • Determinant and volumn change
   • Computation of Determinant
     • Row/Column operations
     • Cofactor expansion
     • Combination of two methods
   • Properties of Determinant
     • Properties
     • Invertibility and determinant
     • Cramer's rule
   • Theory of Determinant
     • Definition of determinant
     • General formula
     • Existence and uniqueness
     • Generalization
   • Proof of Properties
     • Transpose
     • Justification of computation
     • Justification of geometry
5. Eigenvalue and Eigenvector
   • Eigenvalue and Eigenvector
     • Definition
     • Computation
     • Diagonalization
   • Complex Diagonalization
     • Complex eigenvalue and eigenvector
     • Complex eigenvalue and eigenvector for real matrix
     • Complex eigenvalue in real diagonalization
   • Application
     • Discrete dynamical system
     • Functions of matrix
     • Linear differential system
   • Properties
     • About single matrix
     • About several matrices
     • About polynomials of matrix
     • Trace and determinant
   • Eigenspace and Multiplicity
     • Direct sum of eigenspaces
     • Algebraic and geometric multiplicities
     • Multiplicity and diagonalization
   • Polynomials of Matrix
     • Cayley-Hamilton theorem
     • Minimal polynomial
     • Multiplicity and diagonalization
   • Appendix: Polynomial
     • Root and multiplicity
     • Representing polynomial by roots
     • Division of polynomials
     • Greatest common divisor
6. Inner Product
   • Inner Product Space
     • Dot product and euclidean geometry of Rⁿ
     • Inner product
     • Inner product on Rⁿ
   • Duality and Adjoint
   • Orthogonality
     • Orthogonal and orthonormal
     • Properties of orthogonality
     • Orthonormal basis and isometry
   • Orthogonal Projection
     • Orthogonal projection
     • Gram-Schmidt process
   • Orthogonal Complement
     • Definition and properties
     • Complementarity principle
   • Complex Inner Product
     • Definition and properties
     • Complementarity principle