In this chapter we study linear operators $$T : V \to V$$ on a finite-dimensional vector space $$V$$. We are interested in understanding when there is a basis $$B$$ for $$V$$ such that the matrix $$M(T)$$ of $$T$$ with respect to $$B$$ has a particularly nice form. In particular, we would like $$M(T)$$ to be either upper triangular or diagonal. This quest leads us to the notions of eigenvalues and eigenvectors of a linear operator, which is one of the most important concepts in Linear Algebra and beyond. For example, quantum mechanics is largely based upon the study of eigenvalues and eigenvectors of operators on finite- and infinite-dimensional vector spaces.