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Linear Algebra pp 100-130 | Cite as

Representation of Linear Transformations

  • Robert J. Valenza
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

We saw in the previous chapter that a matrix in Mat m×n (k) acts by left multiplication as a linear transformation from k n to km. In this chapter we shall see that in a strong sense every linear transformation of finite-dimensional vector spaces over k may be thus realized. (We say that the associated matrix represents the transformation.) In passing, we introduce the notion of a k-algebra, a rich structure that is a hybrid of both vector space and ring. We show that the set of linear transformations from an n-dimensional vector space to itself is in fact isomorphic as a k-algebra to the familiar matrix algebra M n (k).

Keywords

Vector Space Linear Transformation Transition Matrix Dual Space Canonical Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Robert J. Valenza
    • 1
  1. 1.Department of MathematicsClaremont McKenna CollegeClaremontUSA

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