Let $W\subset(\mathbb{R}^3)^*$ be a subspace formed by the functionals $f:\mathbb{R^3}\rightarrow \mathbb{R}$ such that $Kerf =[(-4,-4,0),(0,2,2)]$.
Find a basis of $W$
The idea I had was: Consider a basis $B=\left\lbrace (-4,-40),(0,2,2)\right\rbrace$ of the $Kerf$ then find a dual basis to $B$ using the Kronecker delta function but the base elements.
I found do not meet the kernel hypothesis.
Any idea or suggestion? Thanks in advance