Linear Algebra pp 100-130 | Cite as

# Representation of Linear Transformations

## 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 *k*^{m}. 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

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