This bachelor's thesis examines topics of Projective Geometry, which will be addressed through Linear Algebra. In particular, the concept of projective space is defined as the set of all one-dimensional vector subspaces of a vector space. Firstly, we introduce the projective transformations through which we prove the two major theorems of Projective Geometry: Desargues’ theorem and Pappus’ theorem. Secondly, we study one of the most important concepts of Projective Geometry: the notion of duality. It is through duality that Projective Geometry can offer a wider range of possibilities if compared to Euclidian Geometry. Thirdly, we define the projective plane curves in a projective plane as well as the homogeneous polynomials so that we can classify the second-degree algebraic curves (conic sections) in a projective plane. Finally, we study the cubic curves with a view to define elliptic curves. The latter can be applied on the elliptic curve cryptosystems, which are the state-of-the art sytems used in modern asymmetric cryptography.