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Let V be the vector space consisting of all polynomials of degree less than or equal to...

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enoterssssssssss | Student, Undergraduate | eNoter

Posted September 4, 2013 at 8:12 AM via web

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Let V be the vector space consisting of all polynomials of degree less than or equal to n with real coefficients. Suppose that `p_(0), p_(1),..., p_(n)` are polynomials in V with the property that `p_(j)(1) = 0` for all j. Prove that the polynomials `p_(0), p_(1),..., p_(n)`  are dependent in V.

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aruv | High School Teacher | Valedictorian

Posted September 4, 2013 at 9:04 AM (Answer #1)

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Let

`P_0(x), P_1 (x),...........P_n (x)`  be polynomials in V.The dimension of V be (n+1).Let polynomial in V are not dependent then these polynomial are linearly independent.Then these polynomials forms a basis of V .But  `P_0 (x)=` constant  ,may not equals to zero say `P_0(x)=2` . `P_0 (x) `  can be written as Linear combination of `P_1(x),......,P_n(x).`

`P_o(x)=a_1P_1(x)+a_2P_2(x)+............+a_nP_n(x)`

`2=0+0+0+....+0`

`` because `P_j(x)=0 `  for x=1  ,which is contradiction . Hence our assumption is wrong. Therefore these polynomials are linearly dependent.

Hence proved.

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