Show that the set is a vector space in the field of real numbers?

V = {p∈ p_3(x)|p(0)=0,p´(0)=0}

1 Answer
KillerBunny
Jun 24, 2018

See below

Explanation:

Let's check all the requirements:

  • The set must be closed with respect to the sum. This is true because, given two polynomials of degree 3, we have

a_0+a_1x+a_2x^2+a_3x^3+b_0+b_1x+b_2x^2+b_3x^3

=(a_0+b_0)+(a_1+b_1)x+(a_2+b_2)x^2+(a_3+b_3)x^3

which is still a polynomial of degree 3. Moreover, if p_0=0 and p_1'(0)=0 and the same goes for p_2, we have

(p_1+p_2)(0)=p_1(0)+p_2(0)=0+0=0

and

(p_1'+p_2')(0)=p_1'(0)+p_2'(0)=0+0=0

  • Closure with respect to scalar multiplication: if p(0)=0 and p'(0)=0, for every k \in \mathbb{R} we still have

(kp)(0)=k\times p(0) = k \times 0 = 0

and similarly

(kp)'(0)=k\times p'(0) = k \times 0 = 0

All of the other requirements (commutative sum, distributive property with respect to the scalars,..) derive from the fact that the sum between polynomials behave as numeric sum, since we just manipolate the like terms as above.

Related questions
Impact of this question
1260 views around the world
You can reuse this answer
Creative Commons License