Abstract
We saw in the previous chapter that a matrix in Mat m×n (k) acts by left multiplication as a linear transformation from kn 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).
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© 1993 Springer Science+Business Media New York
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Valenza, R.J. (1993). Representation of Linear Transformations. In: Linear Algebra. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0901-0_6
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DOI: https://doi.org/10.1007/978-1-4612-0901-0_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6940-3
Online ISBN: 978-1-4612-0901-0
eBook Packages: Springer Book Archive