Let $V$ be the vector space $P_3[x]$ over $\mathbb{R}$. Let $W$ be the span of $\{f_1, f_2\}$, where $f_1 = x^3 − 2x^2 + x$ and $f_2 = x^2 − x$. Find $W^o$.
Since $W$ is the span of the the above mentioned polynomials I took at the the following polynomial
$c_1f_1+c_2f_2=c_1x^3+(-2c_1+c_2)x^2+(c_1-c_2)x$
If I'm not mistaken, this shows that $c_1,c_2=0$. Is this saying that the annihilator is just the zero polynomial: $f(x)=0$?
On
Complete the set $\{f_1, f_2\}$ to a basis of $P_3[x]$: $\{ f_1,f_2,v_1,v_2 \}$
Every linear functional $F$ on the space can be written as:
$F\Big(a_1\cdot f_1+ a_2 \cdot f_2+ a_3 \cdot v_1 + a_4 \cdot v_2 \Big)= a_1 \cdot F(f_1)+ a_2 \cdot F(f_2) + a_3 \cdot F(v_1) +a_4 \cdot F(v_2) $
A functional is in the annihilator if and only if $F(f_1)=0$ and $F(f_2)=0$ (The definition is that $F\vert_W=0$, but this is equivalent).
Hint.
Consider the functionals $$P\longmapsto P(0) \qquad \text{and} \qquad P\longmapsto P(1)$$ Do they vanish on $W$? Are they linearly independent? What should be the dimension of $W^0$?
Note that in coordinates, $P\mapsto P(0)$ is just $(a_3,a_2,a_1,a_0)\mapsto a_0$ and $P\mapsto P(1)$ is just $(a_3,a_2,a_1,a_0)\mapsto a_3+a_2+a_1+a_0$.