Mathematics A2c

Linear algebra:Linear space, subspace, generating system, basis, orthogonal and orthonormal basis.

Arithmetic and rank of matrices.

Determinant: geometric interpretation, expansion of determinants.

Solving systems of linear equations: elementary row operations, Gauss-Jordan- and Gaussian elimination. Homogeneous systems of linear equations.

Cramer’s rule.

Linear maps, linear transformations and their matrices. Kernel, image, dimension theorem.

Linear transformations. Eigenvalues, eigenvectors, similarity, diagonalizability.

Differential equations (DE):

Main definitions: ordinary DE-s, partial DE-s, order of a DE.

Isoclines, slope field.

Separable DEs, first order linear DEs, higher order linear DEs of constant coefficients.

Laplace transforms.

Infinite numerical series:

Convergence, divergence, absolute convergence.

Sequences and series of functions:

Convergence criteria, power series, Taylor series. Fourier series: expansion, odd and even functions.

Multivariable functions:

Continuity, differential and integral calculus, partial derivatives, Young’s theorem. Local and global maxima/minima. Vector-vector functions, their derivatives, Jacobi matrix. Integrals: area and volume integrals.