Linear algebra is the backbone of most of applied mathematics and underlies many areas of physics, such as quantum mechanics. This chapter introduces some of the basics of linear algebra for finite dimensional vector spaces. The properties three-dimensional vectors are generalized to spaces of more than three dimensions in linear algebra courses. The dot product is useful for determining the length of a vector, the angle between two vectors, if the vectors are perpendicular, or projections of one vector onto another. The chapter examines linear systems of differential equations in the plane. A main theme in linear algebra is to study linear transformations between vector spaces. These come in many forms and there are an abundance of applications in physics. Diagonalization is simplest for finite dimensional vector spaces and requires some generalization for infinite dimensional vectors spaces.