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Are linearly independent if no linear combination of distinct Vu V2,... with nonzero coefficients is equal to the zero vector. Thus, the set {0} which is a nonempty subset of every vector space is linearly dependent. On the other hand, every one-element subset of a vector space which is not equal to {0} is linearly independent. Also, we have REMARK 1. If y is a subset of a vector space such that 0 e 9 then 9 is a linearly dependent set of vectors. In view of Definition 2 a necessary and sufficient condition for a non empty subset 9 of a vector space to be linearly independent is that 2 ^ =0 ifandonlyifj f = 0 (11) with Vie 9 and V{ 4= V3 for i 4= j .

Un be subspaces sional vector space V. Then of a finite dimen- r = %+%+> - + °un if and only ifT = °UX + °U2 + (- °lln and dim T = dim °UX H- dim ^ 2 + h dim ^rn. Based on the notion of isomorphism introduced on page 46 we prove the following theorem. T H E O R E M 8. Let °U and T be finite dimensional vector spaces over the same field. , W = T if and only if dim^ = dimr Proof Let °U = T and

1 and let {Uu U2> .

Prove that there are infinitely many elements qofQ such that q2 + 1 = 0. 15. Let SF be a field. Prove that a nonzero ideal of ^\X\ is a prime ideal of $F\X\ if and only if it is generated by a monic irreducible element of 3F\X\. Prove also that every nontrivial prime ideal of 2F\X\ is a maximal ideal of£F\X\. 16. Prove that in a finite field with n elements every element is a root of the polynomial Xn — X. 17. Let 3F be a finite field with n elements and p be an irreducible element of degree m o f f [X].