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I'm wondering whether this question is wrong...

It says show that $\operatorname{orb}(p)$ where $p=[1,1]$ is the set $\{[x,y]: xy\neq0\}$ but if we apply the matrix $$\begin{bmatrix}1&0\\1&-1\end{bmatrix}$$

we get the vector $[1,0]$ which has $xy=0$.

Is it me or the book who is wrong or am I missing the point completely?

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    The orbit of a nonzero vector under GL is the set of nonzero vectors. Maybe they meant $x^2+y^2 \neq 0$ or $x,y \neq 0$.2012-07-05

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