1. Complex numbers
· Numbers sets
· Necessity of complex numbers
· Binomial form of a complex number
· Graphical representation
· Operations
· Complex conjugate, modulus, argument
· Polar form of a complex number
· Roots of complex numbers
· Exponential of a complex number
· Solving equations
2. Systems of linear equations
· Introduction to Linear Equations
· Geometrical Interpretation
· Existence and Uniqueness
· Matrix Notation
· Gaussian Elimination
· Row Equivalence and Echelon Forms
· Solving Linear Systems
· Homogeneous Systems
· Simultaneous Solving
· Systems with parameters
3. The vector space Cn
· Vectors
· Linear Subspace
· Linear Combinations
· Subspace Spanned by Vectors
· Column and Row Spaces
· The Matrix Equation Ax=b
· Null Space
· Revisiting Linear Systems
· Linear Independence
· Basis for a Linear Subspace
· Dimension of a Linear Subspace
· Basis for Col A, Row A and Nul A
· Rank of a Matrix
· Coordinate Systems
· Introduction to Linear Transformations
4. Matrix algebra
· Matrix Operations
· Transpose of a Matrix
· Conjugate Transpose of a Matrix
· Inverse of a Matrix
· Partitioned Matrices
· Determinants
5. Eigenvalues and eigenvectors
· Eigenvalues & Eigenvectors
· The Characteristic Equation
· Diagonalization
· Change of Basis
· Transformations between Linear Subspaces
· Applications to linear systems of differential equations
6. Orthogonality
· Dot Product and Modulus
· Orthogonal Sets
· Unitary Matrices
· Orthogonal Complement
· Orthogonal Projection
· The Gram-Schmidt Process
· The QR decomposition
· Least-Squares Problems
7. Normal matrices
· Schur Decomposition
· Normal Matrices & Unitary Diagonalization
· Particular Cases of Normal Matrices