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Question: please help me with detailed answers thank you ...

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Problem 4 (2 points). Let F be a field with additive identity 0 and multi- plicative identity 1. Let S be the space of eventually zero infinite sequences of members of F, i.e. the space of infinite sequences of members of F that have only finitely many non-zero entries.4 For all i E N, let s, be the sequence whose i-th term is 1, and all of whose other terms are zero, i.e (Q.0 1,0,0,0..) Prove that (81, 82, 83,...J is a basis for S. In other words, prove that the following hold: (a) [I points/ every non-empty finite subset of (s inearly independent; (b) [1 points] every s e S is a linear combination of finitely many members of (81, 82, 83,....]. 2Hint: First prove that (po(x),pi(x),... ,pn(x) is a linearly independent set in Pn, and then use the fact that dim(pn) n+1. 3Hint: First, express the desired p(x) as a linear combination of po(x),pi(z),... pm (x). Then, prove that if q(x) is any other linear combination of po(x),pi(ax),. Pn(x), then q(x) does not have the desired property Here, the field of scalars is F. Vector addition and scalar multiplication are defined in the natural way, i.e. term by term. You may assume that S is indeed a vector space over F. Please help me with detailed answers, thank you! :)
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