Vector spaces and subspaces, generators, linearly independent sets, bases, dimension. Sum and intersection of subspaces, direct sums.
Matrices, matrix operations and their properties, column space, row space, and null space, rank, row reduction and echelon form, determinant, invertible matrices.
Solvability and solution sets of linear systems, affine subspaces.
Linear maps, kernel, image, dimension formula, injectivity and surjectivity. Matrices of a linear map, basis change.
Endomorphisms, eigenvectors, eigenvalues and diagonalizability.
Bilinear forms, scalar products, euclidean spaces, orthogonality, orthogonal projections, Gram-Schimdt process, orthogonal decomposition. Diagonalizability of real symmetric matrices. Quadratic forms, canonical form and normal form.
Vectors in 3-space, dot product, cross product, triple product, angle between two vectors, orthogonal projections, parallelogram area and parallelepiped volume. Parametric and cartesian equations of lines and planes, lines and planes intersections, parallelism, skew lines, distances
to points, lines, planes.
Conics, canonical forms, matrix representation and classification, plane rotations, completing the square method.
Canonical equations of the quadrics.