Abstract
We are familiar with the properties of finite dimensional vector spaces over a field. Many of the results that are valid in finite dimensional vector spaces can very well be extended to infinite dimensional cases sometimes with slight modifications in definitions. But there are certain results that do not hold in infinite dimensional cases. Here we consolidate some of those results and present it in a readable form.The linear transformation on infinite dimensional vector spaces and introduce the concept of infinite matrices. We will show that every linear transformation corresponds to a row finite matrix over the underlying field and vice versa and will prove that the set of all linear transformations of an infinite dimensional vector space in to another is isomorphic to the space of all row finite matrices over the underlying field.
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