Let $p=(0,0), q=(0,1), r=(1,0)$ be points of $\mathbb{C}^2$. What is the dimension of the $\mathbb{C}$-vector space $\{f(X,Y) \in \mathbb{C}[X,Y] \ | \text{ deg}f \leq 2 \text{ and } f(p)=f(q)=f(r)=0\}$, where by deg$f$, we mean the total degree of the polynomial $f$?
What does $\mathbb{C}$-vector space mean? Can anyone give me a hint for proceeding
A polynomial in two variables of degree $\le 2$ has the form: $$f(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F$$ You are given that for the polynomials $f$ in your vector space, $$f(0,0) = 0\\f(0,1) = 0\\f(1,0) = 0$$ That gives you 3 equations in the $6$ unknowns $A, B, C, D, E, F$.
So when you've solved those three equations as far as you can, how many independent unknowns will you have left?