The exam topics

Linear algebra and analytic geometry

Vectors, matrices and linear systems, vector spaces and linear maps, eigenvalues and diagonalization, lines, planes, conics and quadrics.

1. Vectors in the plane and in space

   What a geometric vector is, how to compute with components, and how the dot product, the cross product and the scalar triple product measure angles, areas and volumes and detect orthogonality, parallelism and coplanarity.

2. Matrices and matrix operations

   Matrices and their types, sum, row-by-column product, transpose and powers, the inverse matrix, row echelon reduction and the computation of the inverse with the Gauss-Jordan method.

3. Determinant and inverse matrix

   How to compute the determinant of a square matrix, which properties simplify the computation, and how the determinant decides invertibility and gives the formula for the inverse.

4. Linear systems and rank

   How to write a linear system in matrix form, how to compute the rank, when a system has solutions and how many, how to solve it with Gauss or with Cramer, and how to discuss it as a parameter varies.

5. Vector spaces, bases and dimension

   The common structure of vectors, matrices and polynomials, subspaces described by generators or by equations, bases, dimension and Grassmann's formula for sum and intersection.

6. Linear maps

   The functions between vector spaces that preserve sums and scalar multiples, the matrix that represents them, kernel and image, the rank-nullity theorem, isomorphisms, composition and change of basis.

7. Eigenvalues, eigenvectors and diagonalization

   How to compute the eigenvalues and eigenspaces of a matrix or of an endomorphism, when a matrix is diagonalizable and how to build the diagonalizing matrix, with applications to powers and to matrices with a parameter.

8. Inner product, orthogonality and the spectral theorem

   The standard scalar product, norm and angles, orthonormal bases, orthogonal projections with the Gram-Schmidt process, orthogonal matrices and the orthogonal diagonalization of symmetric matrices.

9. Lines and planes in space

   Equations of planes and lines in space, relative positions, angles, distances, orthogonal projections, pencils of planes, and the line of minimum distance between two skew lines.

10. Conics and quadrics

    How to write the matrix of a conic or a quadric, how to classify it with the invariants, and how to reduce it to canonical form with a rotation and a translation, using eigenvalues and eigenvectors.