4 Credits (3-1-0)

Revision of existence-uniqueness of solutions of a system of linear equations, elementary row operations, row-reduced echelon matrices.

Vector spaces, span of a subset, bases and dimension, quotient spaces, direct sums. Linear transformations, rank-nullity, matrix representation of a linear transformation, algebra of linear transformations, dual space, transpose of a linear transformation. Eigenvalues, eigenvectors, annihilating polynomials, Cayley-Hamilton theorem, invariant subspaces, triangulable and diagonalizable linear operators. Simultaneous triangulation and diagonalization, Primary decomposition theorem, Jordan decomposition.

Inner product spaces over R (real numbers) and C (complex numbers), Gram-Schmidt orthogonalization process, orthogonal projection, best approximation. Adjoint of a linear operator, unitary and normal operators, spectral theory of normal operators. Bilinear forms, symmetric and skew-symmetric bilinear forms.