Mathematics for Physics I

Complex numbers. Definition. Algebraic properties. Geometric representation, modulus, and the conjugate of a complex number. The triangle inequality. Polar form: argument and principal value of the argument of a complex number. Exponential form: Euler's formula. Powers and roots: de Moivre's formula. 

Matrices and Gaussian elimination. The geometry of linear equations. An example of Gaussian elimination. Matrix notation and matrix multiplication. Triangular factors and row exchanges: LU factorization. Inverse matrices and transpose matrices.

Vector spaces and linear equations. Vector spaces and subspaces. Elimination in a linear system of m equations with n unknowns: the solutions of Ax=0 and the solutions of Ax=b. Linear independence, basis, and dimension. The four fundamental subspaces. Matrices and linear maps (linear transformations).

Orthogonality. Orthogonal vectors and orthogonal subspaces. Inner products and projections onto lines. Projections and the optimal least squares solutions of Ax=b. Orthogonal bases, orthogonal matrices, and the Gram-Schmidt orthonormalization.

Determinants. Properties of the determinant. Formulas for the determinant. Applications of determinants: computation of the inverse matrix, Cramer's rule, the volume of an n-dimensional parallelepiped, a formula for the pivots.

Eigenvalues and eigenvectors. The solutions of Ax=λx. Diagonalization of a matrix. Difference equations and powers Ak. Complex matrices: symmetric vs Hermitian and orthogonal vs unitary. Similarity transformations.