Linear algebra is a difficult topic for undergraduate students. In France, the focus of beginning linear algebra courses is the study of abstract vector spaces, with or without an inner product, rather than matrix operations as is common in many other countries. This paper presents a study of the possible uses of geometry and "geometrical intuition" in the teaching and learning of linear algebra. Fischbein's work on intuition in science and mathematics is used to analyze the treatment and use of geometry in linear algebra textbooks as well as mathematicians' and students' uses of geometry in linear algebra. I indicate the possibilities and limitations of such uses of geometry and make suggestions for a linear algebra course that uses geometry to support learning.