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
linear mappings.
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.