V is an algebra of linear transformations, where in this algebra the product AB of two linear transformations A, B E 'c(V) is defined to be their composition; that is, AB is the linear transformation on V that acts on each vector v E V according to the rule (AB)v = A(Bv).

1. There exists a field extension iF of IF such that (i) iF is an algebraic extension of IF, and (ii) iF is algebraically closed. Moreover, if IT{ is any algebraically closed algebraic extension of IF, then there is a field isomorphism T : IT{ ---t iF such that T(() = ( for all ( E IF. 2. If IT{ is a field extension of JR. , then IT{ and C are isomorphic fields. 6 Existence of Bases for Infinite-Dimensional Spaces This section is not a prerequisite for reading the subsequent chapters. There are many advantages to a development of linear algebra that does not presuppose that the vector spaces under study have finite dimension.

T",) - A (~ , T",) . In other words , ~~
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