Course information

The main object of the course is to understand the basic concepts of Linear Algebra and Analytical Geometry. Course Contents included: Vectors in space, coordinates and vector operations, inner product, outer product and mixed product of vectors. Line in space (vector equation, parametric equations, analytic equations), relative positions of lines. Plane in space (vector equation, parametric equations, analytic equation), relative positions of planes. Matrices, matrix operations, properties of operations and identities, inverse matrix. Determinants, properties of determinants, adjoint matrix, calculating the inverse matrix using determinants. Linear systems, scaling a matrix with elementary row-transformations and solving linear systems using the Gauss elimination method, calculating the inverse matrix using the Gauss-Jordan method, solving linear systems using the Cramer method. Vector spaces, vector subspaces, sum and intersection of subspaces, linear hulls, linear independence and dependence of vectors, basis and dimension, dimensions’ theorem. Inner product spaces, vector norm, Cauchy-Schwarz inequality, parallelogram rule. Gram-Schmidt orthonormalization, unitary matrices, QR factorization, least squares problem in a linear system, pseudoinverse matrix. Eigenvalues and eigenvectors of matrices, properties of eigenvalues and eigenvectors, characteristic polynomial. Diagonalization of a matrix, Cayley-Hamilton theorem, minimum polynomial. Diagonalization of symmetric real matrices, quadratic forms.