In these lectures we shall see how one can use linear algebra to count certain combinatorial quantities. The student may be familiar from basic graph theory that to each graph is associated its adjacency matrix P, and to count the number of paths of length k between two vertices i and j, we simply read off the ijth entry of P^k.
While this is a simplistic model, we shall see that it contains the basic idea of the method- associate a linear algebraic object (matrices, vector spaces etc) to the combinatorial quantity and derive bounds by linear algebra (rank, linear indepenfence etc). We will familiarize ourselves with this method by studying concrete examples such as constructive Ramsey theory, two distance sets, and other combinatorial objects arising in geometry. A basic familiarity with linear algebra (rank, linear independence etc) is the only prerequisite.