Determine if each of the following sets is a subspace of PP (the vector space of

Question

Determine if each of the following sets is a subspace of PP (the vector space of polynomials). Type "yes" or "no" for each answer.

Let W1W1 be the set of all polynomials of the form p(t)=at2p(t)=at2, where aa is in RR.  

Let W2W2 be the set of all polynomials of the form p(t)=t2+ap(t)=t2+a, where aa is in RR.  

Let W3W3 be the set of all polynomials of the form p(t)=at2+atp(t)=at2+at, where aa is in RR.

(2 points) Determine if each of the following sets is a subspace of P (the vector space of polynomials). Type "yes" or "no" for each answer. Let W1 be the set of all polynomials of the form p(t) at2, where a is in R. Let W2 be the set of all polynomials of the form p(t) = t2 + a, where a is in R Let W3 be the set of all polynomials of the form p(t) = at2 + at, where a is in R. fall polynomials of the form pt) -+ a, where a is in R.

Explanation / Answer

(a) p(t) = at2

Consider p(t)+q(t) = at2 +bt2 = (a+b)t2 is in W1

Also bp(t) = b(at2) = (ba)t2 is in W1 [because ab is real number]

Hence W1 is a subspace ...Yes

(b) Consider p(t)+q(t) = (t2+a)+(t2+b) = 2t2+(a+b) is not in W2

Hence W2 is not a subspace...No

(c) p(t)+q(t) = (at2+at)+(bt2+bt) = (at2+bt2)+(at+bt) = (a+b)t2+(a+b)t is in W3

Consider bp(t) = b(at2+at) = (ba)t2+(ba)t is in W3

Hence W3 is a subspace...Yes

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