Book contents
- Frontmatter
- Preface
- Contents
- To the binary tree
- 1 Vector spaces and linear transformations
- 2 Coordinates
- 3 Determinants
- 4 The structure of a linear transformation I
- 5 The structure of a linear transformation II
- 6 Bilinear, sesquilinear, and quadratic forms
- 7 Real and complex inner product spaces
- 8 Matrix groups as Lie groups
- A Polynomials
- B Modules over principal ideal domains
- Bibliography
- Index
- About the Author
4 - The structure of a linear transformation I
- Frontmatter
- Preface
- Contents
- To the binary tree
- 1 Vector spaces and linear transformations
- 2 Coordinates
- 3 Determinants
- 4 The structure of a linear transformation I
- 5 The structure of a linear transformation II
- 6 Bilinear, sesquilinear, and quadratic forms
- 7 Real and complex inner product spaces
- 8 Matrix groups as Lie groups
- A Polynomials
- B Modules over principal ideal domains
- Bibliography
- Index
- About the Author
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.
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- Information
- A Guide to Advanced Linear Algebra , pp. 89 - 108Publisher: Mathematical Association of AmericaPrint publication year: 2011