4 - The structure of a linear transformation I

Summary

In this chapter we begin our analysis of the structure of a linear transformation J : V → V, where V is a finite-dimensional F-vector space.

We have arranged our exposition in order to bring some of the most important concepts to the fore first. Thus we begin with the notions of eigenvalues and eigenvectors, and we introduce the characteristic and minimum polynomials of a linear transformation early in this chapter as well. In this way we can get to some of the most important structural results, including results on diagonalizability and the Cayley-Hamilton theorem, as quickly as possible.

Recall our metaphor of coordinates as a language in which to speak about vectors and linear transformations. Consider a linear transformation J : V → V, V a finite-dimensional vector space. Once we choose a basis B of V, i.e., a language, we have the coordinate vector [v]_B of every vector v in V, a vector in Fⁿ, and the matrix [J]_B of the linear transformation J, an n-by-n matrix, (where n is the dimension of V) with the property that |J(v)]_B = [J]_B[v]_B. If we choose a different basis e, i.e., a different language, we get different coordinate vectors [v]_e and a different matrix [J]_e of J, though again we have the identity |J(v)]. We have also seen change of basis matrices, which tell us how to translate between languages.