determine whether the est of all polynomials P(x) of degré no more than two for which...

determine whether the est of all polynomials P(x) of degré no more than two for which p(0)=1, with regular addition of polynomials and regular multiplication by a number, forms a vector space

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Expert Solution

The set pf polynomials of degree no more than two for which P(0)=1,with regular addition of polynomials and regular multiplication by a number doesnot form a vector space. Reason is as follows

The very five basic requirements of a vector space are

1) sum of any two vwctors in this space should be another vector in the space.

This is not satisfied by the given set. Let P(x) be a memeber of the set then -P(x) will also the membeher of the set. Now P(x)+(-P(x))=0.

Zero canot be a member of the set because of the restriction P(0)=1

2) Vector addition should be commutative

3) Vector addition should be associative

4) There should be a zero vector such that V+ Zerovector=V

5)There should be an an inverse vector (-V) for every vector V, such that V+(-V)=Zero vector

The conditions 2 to 5 are also not satisfied by the given set due to same arguments of 1. Therefore the given set doesnot form a vector space

genius_generous answered 1 year ago

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