Let H be the set of all polynomials of the form pt a bt2 w

Let H be the set of all polynomials of the form p(t) = a + bt^2 where a and b are in R and b Greaterthan a Determine whether H is a vector space If it is not a vector space determine which of the following properties it fails to satisfy. Contains a zero vector Closed under vector addition Closed under multiplication by scalars H is not a vector space because it is not closed under multiplication by scalars and does not contain a zero vector H is not a vector space because it does not contain a zero vector H is not a vector space because it is not closed under vector addition. H is a vector space

Solution

1.

Set a=0 and b=0 and we see it contains a zero vector

2.

Let,p=a+bt^2,q=c+dt^2 belong to H

p+q=a+c+(b+d)t^2 which belongs to H

Hence closed under vector addiction

3.

Let c be a real number and, p=a+bt^2 belong to H

cp=ca+cbt^2

Hence closed under multiplication by scalars

Hence D. H is avector space