We will introduce the concept of a Lie algebra by defining a new product on matrices called the commutator which has several important properties that make it distinct from matrix multiplication. In particular, we will see that associativity is replaced by the Jacobi identity. This new algebra, called the general linear algebra, has several important subalgebras such as the special linear subalgebra of matrices with trace zero. In the one and two dimensional cases, we will investigate how these Lie algebras naturally act on vector spaces, and discuss Lie algebra representations. If time allows, we will see how we can use the commutator to turn first order differential operators into a Lie algebra. The only prerequisite for this talk is an understanding of undergraduate linear algebra and basic calculus.