Determine if the following is a subspace a Is the given set

Determine if the following is a subspace

(a) Is the given set a subspace of P₂? Justify your answer. H={p(t)=4a +at+2at2 where thea_i are any real numbers}

Solution

p(t)=4a+at+2at^2

1. 0 is in H for a=0

2. Let, p,q be in H

So there exist a,b so that:

p(t)=4a+at+2at^2

q(t)=4b+bt+2bt^2

p+q=4a+at+2at^2+4b+bt+2bt^2=4(a+b)+(a+b)t+2(a+b)t^2

3.

Let, p(t)=4a+at+2at^2 be in H and c be a real number

cp(t)=4ca+cat+2cat^2 is also in H

Hence, H is a subspace.