A subvariety of an affine algebraic variety $V\subseteq\mathbb{C}^n$ is an affine algebraic variety $W\subseteq\mathbb{C}^n$ that is contained in $V$.
So with respect to this definition, is it true that the set $U(n)$ of all unitary matrices is not an affine algebraic subvariety of $\mathbb{C}^{n^2}$?
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