In: Physics
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
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