1. Complex numbers
- Number sets
- The need for complex numbers
- Binomial form of complex numbers
- Graphical representation
- Operations
- Conjugate, module and 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 systems
- Geometric interpretation
- Existence and uniqueness
- Matrix notation
- Gaussian elimination
- Equivalence by rows, echelon form
- Resolution of linear systems
- Homogeneous systems
- Simultaneous resolution
- Systems with parameters
3. The vector space Cn
- Vectors
- Vector subspaces
- Linear combinations
- Subspace spanned by a set
- Column and row spaces
- The matrix equation Ax=b
- Null space
- Revisiting linear systems
- Linear independence
- Base of a vector subspace
- Dimension of a vector subspace
- Bases of Col A, Row A and Nul A
- Rank of a matrix
- Coordinate systems
- Introduction to linear transformations
4. Matrix algebra
- Operations with matrices
- Transposition of a matrix
- Conjugated transposition of a matrix
- Inverse of a matrix
- Block matrices
- Determinants
5. Eigenvalues and eigenvectors
- Eigenvectors and eigenvalues
- The characteristic equation
- Diagonalization
- Change of basis
- Linear transformations between vector spaces
- Abstract vector spaces
6. Orthogonality
- Scalar product and module
- Orthogonal sets
- Unitary matrices
- Orthogonal complement
- Orthogonal projections
- The Gram-Schmidt process
- Least squares problems
- Singular value decomposition