This is a unified treatment of the various algebraic approaches to geometric spaces‎.‎ The study of algebraic curves in the complex projective plane is the natural link between linear geometry at an undergraduate level and algebraic geometry at a graduate level‎,‎ and it is also an important topic in geometric applications‎,‎ such as cryptography‎.‎ 380 years ago‎,‎ the work of Fermat and Descartes led us to study geometric problems using coordinates and equations‎.‎ Today‎,‎ this is the most popular way of handling geometrical problems‎.‎ Linear algebra provides an efficient tool for studying all the first degree ‎(‎lines‎,‎ planes‎)‎ and second degree ‎(‎ellipses‎,‎ hyperboloids‎)‎ geometric figures‎,‎ in the affine‎,‎ the Euclidean‎,‎ the Hermitian and the projective contexts‎.‎ But recent applications of mathematics‎,‎ like cryptography‎,‎ need these notions not only in real or complex cases‎,‎ but also in more general settings‎,‎ like in spaces constructed on finite fields‎.‎ And of course‎,‎ why not also turn our attention to geometric figures of higher degrees‎?‎ Besides all the linear aspects of geometry in their most general setting‎,‎ this book also describes useful algebraic tools for studying curves of arbitrary degree and investigates results as advanced as the Bezout theorem‎,‎ the Cramer paradox‎,‎ topological group of a cubic‎,‎ rational curves etc‎.‎ Hence the book is of interest for all those who have to teach or study linear geometry‎:‎ affine‎,‎ Euclidean‎,‎ Hermitian‎,‎ projective‎;‎ it is also of great interest to those who do not want to restrict themselves to the undergraduate level of geometric figures of degree one or two‎.‎