Linear Algebra 2400-FIM1AL

1 Systems of linear equations: solutions and general solutions, matrices, elementary matrix operations, solving the system of equations using Gaussian elimination.

2 Linear (or vector) spaces: examples, linear subspaces, linear combinations of vectors, linear independence, basis and dimension of a linear space, the coordinates of the vector in a given basis.

3 Linear transformations: examples , matrix representation of linear transformations, the algebra of linear transformations and matrix operations, matrix algebra .

4 Determinants: properties of determinants and methods of calculation.

5 Matrix Inverse and methods of finding the inverse matrix.

6 Applications determinant and rank of a matrix to solve linear equations : Kronecker - Capelli theorem and Cramer.

7 Vectors and eigenvalues ​​of linear transformations: Find the eigenvalues, the characteristic polynomial, bases of eigenspaces and diagonalisation of matrices.

8 Applications matrix diagonalisation.

9 Affine subspaces (or layers) of linear spaces, equations of the line and plane.

10 The standard scalar product: vector length, magnitude of vectors, orthogonal bases and orthonormal bases and the Gram-Schmidt procedure.

11 Quadratic forms: examples of matrix quadratic forms, Sylvester criterion of positive definiteness and tests of semidefinitness using eigenvalves.