Representation of Linear Transformations
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).
KeywordsVector Space Linear Transformation Transition Matrix Dual Space Canonical Basis
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