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TIMESTAMPS
The Wayback Machine - https://web.archive.org/web/20090525025830/http://algebra.math.ust.hk:80/course/content.shtml
Linear Algebra
Table of Contents
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
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
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
*
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
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
Inner Product
Inner Product Space
Dot product and euclidean geometry of
R
n
Inner product
Inner product on
R
n
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
Linear Algebra
