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Definition An affine map from $F^n$ to $F^m$ is a function $f : F^n \to F^m$ given by $f(x)=Ax+b$ for some $m \times n$ matrix $A$ with entries in $F$ and some vector $b \in F^m$.

Definition An algebraic set $X \subseteq F^n$ is affinely equivalent to an algebraic set $Y \subseteq F^m$ if there is a bijective affine map $f: X \to Y$ whose inverse is also an affine map $Y \to X$.

Question Show that $X=V(4xy+8xz+4yz-3y^2)$ is affinely equivalent to $Y=V(x^2+y^2-z^2)$ in $\mathbb{R}^3$.

My Attempt: I know the key point is to express the relation $4xy+8xz+4yz-3y^2$ into the form $u^2+v^2-w^2$. This question should just be a question of mathematical tricks. I have tried to expand the relation $(ax+by)^2+(cx+dz)^2-(ey+fz)^2$ then match the coefficients with $4xy+8xz+4yz-3y^2$. But I failed.

Then I tried to expand $(ax+by+cz)^2+(dx+ey+fz)^2-(gx+hy+iz)^2$ and match the coefficients with $4xy+8xz+4yz-3y^2$. But still failed.

What should I do? This should be a question about tricks but I don't know the trick to rewrite the expression $4xy+8xz+4yz-3y^2$. I have been stuck on this question for two days...Can anyone help me?

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Hint: The rank-3 real quadratic form $4xy+8xz+4yz-3y^2$ has signature $-1$, not $+1$, so match it with $-u^2-v^2+w^2$ instead.

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  • $\begingroup$ Thank you ! By the way, what is a signature? Can you give a reference for this concept? I didn't learn any "signature thing" in my linear algebra course... $\endgroup$ – bbw Jun 11 at 5:19
  • $\begingroup$ See Sylvester's law of inertia. $\endgroup$ – user10354138 Jun 11 at 5:24

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