[Math] Linear Algebra – Dual vector Space

linear algebra

My question is:

''In this problem, we shall explore some if the concepts surrounding dual vector spaces for the specific case of $V= P(3)$, the vector space of polynomials with real coeficients of degree less than or equal to $3$. Assume through that $V=P(3)$

a) Describe a basis for $V^*$ (dual vector space) and express the linear functional $f:V\to R$ given by: $$f(a_3x^3+a_2x^2+a_1x+a_0)=2a_3+a_1-5a_0$$ as a linear combination of the basis you found for $V^*$.

b) Find a basis for the annihilator $W$ of the subspace $W= \text{span}(1+x^2, x+x^3)$ which belongs to $C$. What's dimension of annihilator of $W$?

c) We know that the evaluation map $E:V\to V^{**}$ is a vector space isomorphism. Then $E(x^2+x^3)$ is a linear functional on $V^*$. What is the value of the linear functional $E(x^2+x^3)$ on the linear functional $f$ from part (a) ? (I.e, what is $E(x^2+x^3)(f)$?''
Please help me ! I even don't know how to start !

Best Answer

I'll give some hints for you to get started:

  1. If $V$ is a finite dimensional vector space with basis $B$, then there's a natural choice for a basis $B^\ast$ in the space $V^\ast$ which is the dual basis how can you use this?
  2. Recall that the annihilator of $ W \subset V$ is the set of all $f \in V^\ast$ such that $f(w)=0$ for all $w \in W$. It seems you know a basis for $W$ agree? So, use the definition!
  3. Write down explicitly the evaluation map and apply it. Example: if we take $x \in P(3)$, then the evaluation $E: V \to V^{\ast\ast}$ is $E(x)(f) = f(x)$.

If you need further aid ask in comment.

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