Linear Algebra 2400-L1PPAL

1. Systems of linear equations: particular solutions and the general solution, Gauss elimination method.
2. Linear ( or vector) spaces: examples, subspaces, linear combination of vectors, linear independence of a system of vectors, basis and dimension of a linear space, coordinates of a vector in a basis.
3. Linear transformations: examples, the matrix of a linear transformation in bases. Operations on linear transformations and operations on matrices. Matrix algebra.
4. Determinants: properties and computation methods.
5. Inverse and transpose matrix, their computation.
6. Matrix rank: its relation to elementary operations and to determinant.
7. Applications of matrix rank in solving of linear systems of equations. Kronecker – Capelli and Cramer theorems.
8. Eigenvalues and eigenvectors of a linear tranformation.
Characteristic polynomial of a matrix. Finding eigenvalues and eigenspaces of a transformation. Diagonal and diagonalizable matrices. Matrix trace and its properties.
9. Applications of matrix diagonalization.
10. Affine spaces and affine tranformations.
11. Standard scalar product: vector norm, orthogonal systems of vectors, orthogonal and orthonormal bases. Gram – Schmidt procedure. Orthogonal projections and symmetries.
12. Quadratic forms: examples, form matrix. Definite form and Sylvester cryterion. Application of eigenvalues to definite and semi-definite forms.
13. Introduction to linear programming. Symplex method.