Fields: definitions, the field of complex numbers.
Linear equations: elementary operations, row reduction, homogeneous and inhomogeneous systems, representations of the solutions.
Vector spaces: examples, subspaces, linear dependence, bases, dimension.
Matrix algebra: matrix addition and multiplication, elemetary operations, the inverse of a matrix, the determinant, Cramer’s rule.
Linear transformations: examples, kernel and image, matrix representation.
Diagonalization: eigenvectors and eigenvalues, the characteristic polynomial, applications.
Bilinear forms.
Finite dimensional inner product spaces.
Operators on finite dimensional inner product spaces: the adjoint, self adjoint operators, normal operators, diagonalization of normal operators.

Linear Algebra for physics students(!)