- The real numbers, inequalities in real numbers, the complex numbers, the Cartesian
representation, the polar representation, the exponential representation, the Theorem of de
Moivre, root computations.
- Systems of linear equations over the real or complex numbers, the solution set and its
parametric representation, echelon form and the reduced echelon form of a matrix, backwards
substitution, forward substitution and their complexity, the Gauss elimination algorithm and its
complexity, the reduction algorithm and its complexity.
- Vector spaces, sub-spaces of vector spaces, linear combinations of vectors, the span of a set
of vectors, linear dependence and linear independence, the dimension of a vector space, row
spaces and column spaces of matrices, the rank of a matrix.
- Linear mappings between vector spaces, invertible mappings and isomorphisms, the matrix
representation of finite dimensional linear mappings, inversion of a square matrix, composition
of mappings, multiplication of matrices, the algebra of matrices, the kernel and the image of a
linear mapping and the computation of bases, changing of a basis, the dimension theorem for
- Inner product spaces, orthogonality, the norm of a vector, orthonormal sets of vectors, the
Cauchy-Schwarz inequality, the orthogonal complement of a sub-space, orthogonal sequences
of vectors, the Gram-Schmidt algorithm, orthogonal transformations and orthogonal matrices.
- The determinant of a square matrix, minors and cofactors, Laplace expansions of the
determinant, the adjoint matrix and Laplace theorem, conjugation of a square matrix, similarity
transformations and their invariants (the determinant and the trace).
- Eigenvalues, eigenvectors, eigenspaces, diagonalization and similarity, the characteristic
polynomial, the algebraic and the geometric multiplicities of an eigenvalue, the spectral
theorem for Hermitian matrices.