Algebra2

This second-semester course in Linear Algebra provides a rigorous exploration of finite-dimensional vector spaces and their structural properties over the real and complex fields. Building on introductory concepts, the curriculum begins with an in-depth study of vector spaces, subspaces, sums and direct sums, linear independence, and bases. It then develops the theory of linear maps including kernel, image, rank, and injectivity/surjectivity criteria and establishes their representation via matrices. Students will master matrix algebra, determinants, and inverse matrices, while learning to connect operations on linear maps with corresponding matrix operations. A central theme is the interplay between coordinate representations and change of basis. The course covers the theory of linear systems, with emphasis on Cramer’s systems, and culminates in the diagonalization and triangularization of matrices via eigenvalues, eigenvectors, characteristic polynomials, and the Cayley-Hamilton theorem. Throughout, theoretical understanding is reinforced through proofs, computational techniques, and applications across mathematics and related scientific disciplines.