Let be the set of all infinite sequences in a field with addition and scalar multiplication defined as below: For x = {x_n }= x_1, x_2, …….,x_n,……..∈ X and y = {y_n }= y_1, y_2,….., ∈ X x + y= {x_n } + {y_n } = x_1 + y_1, x_2 + y_2,…….., x_n + y_n,….and cx=c{x_n }=cx_1, cx_2, cx_3,……cx_n,… where x_n, y_n and c are all in F, n=1,2,3……. Show that X is a vector space over F.

Let be the set of all infinite sequences in a field with addition and scalar multiplication defined as below: For x = {x_n }= x_1, x_2, …….,x_n,……..∈ X and y = {y_n }= y_1, y_2,….., ∈ X x + y= {x_n } + {y_n } = x_1 + y_1, x_2 + y_2,…….., x_n + y_n,….and cx=c{x_n }=cx_1, cx_2, cx_3,……cx_n,… where x_n, y_n and c are all in F, n=1,2,3……. Show that X is a vector space over F.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.2: Divisibility And Greatest Common Divisor

Problem 35E

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