1. Complex numbers
1. Definitions
2. Forms of complex numbers
3. Properties and operations
2. Matrices
1. Operations with matrices
2. Transpose and inverse of a matrix
3. Determinants
4. Sets associated to a matrix
3. Systems of linear equations
1. Introduction and definitions
2. Geometric interpretation
4. Matrix methods to solve linear systems: Gauss and Gauss-Jordan. Application to the inverse of a square matrix
5. Homogeneous systems
4. Vector Spaces
1. Definitions
2. Operations and properties
3. Vector subspaces
4. Spanning sets
5. Linear dependence and independence
6. Bases and dimension. Coordinates. Change of bases. Change of coordinates.
5. Linear Transformations
1. Definition, properties and operations
2. Inverse of a linear transformation
3. Kernel and range of a linear transformation
6. Linear transformations and matrices
1. Representation of linear transformations from IR^m to IR^n with matrices
2. Representation of linear transformations between arbitrary vector spaces
7. Eigenvalues and eigenvectors of a square matrix
1. Definitions
2. Similarity and diagonalisation
3. Spectral theorem
8. Orthogonality
1. Definitions. Inner product. Length of a vector. Angle between two vectors. Orthogonal projection
2. Orthogonal and orthonormal bases
3. Orthogonal matrices and orthogonal linear transformations
4. Orthogonal subspaces and orthogonal complement
5. The Gram-Schmidt process and the QR factorisation
9. Least squares
1. Best approximation in the sense of least squares
2. Computation of the least squares solution
3. Applications to data fit and approximation of functions with polynomials
10. Introduction to Linear Ordinary Differential Equations with constant coefficients
9.1. Introduction to continuous dynamical systems and differential equations
9.2. Linear ordinary differential equations
9.3. Solution to systems of linear ordinary differential equations with constant coefficients
9.4. Introduction to stability of continuous dynamical systems
0. Review Topics
0.1. Introduction to Linear Systems
0.2. Basics vectors and matrix
Complex numbers
1. Definitions
2. Forms of complex numbers
3. Properties and operations
1. Systems of linear equations
1.1. Introduction and definitions
1.2. Geometrical interpretation
1.3. Techniques for solving linear systems
1.4. Matrix methods: Gauss and Gauss-Jordan
1.5. Homogeneous linear system
2. Vector spaces
2.1. Definitions
2.2. Operations and properties
2.3. Vector subspaces
2.4. Linear combinations and Span
2.5. Linear independence
2.6. Bases and dimension of a subspace
2.7. Dot product. Length of a vector. Angle between two vectors
2.8. Orthogonal projection
3. Matrices
3.1. Operations with matrices
3.2. Transpose and inverse of a matrix
3.3. Determinant
3.4. Matrix subspaces
4. Linear transformations
4.1 Definitions, properties and operations
4.2. Inverse of a linear transformation
4.3. Image and kernel of a linear transformation
5. Bases
5.1. Coordinates
5.2. Change of basis
6. Orthogonality
6.1. Definitions
6.2. Orthogonal and orthonormal bases
6.3. Orthogonal matrix and linear transformations
6.4. Orthogonal projections and orthogonal complements
6.5. Gram-Schmidt process and QR factorization
7. Least squares
7.1. Better approximation.
7.2. Approximation using least squares
7.3. Methods and applications in data fitting and approximation of functions by polynomials
8. Eigenvalues and eigenvectors
8.1. Definitions
8.3. Similarity and Diagonalization
8.4. Spectral theorem
9. Introduction to Linear Ordinary Differential Equations with Constant Coefficients
9.1. Introduction to Continuous Dynamical Systems and Differential Equations
9.2. Linear Ordinary Differential Equations
9.3. Linear systems of differentiqal equations with constant coefficients
9.4. Introduction to the Stability of dynamical systems