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Let U = $\operatorname{span}\{(1,0,0,-1),(0,1,-1,0)\}$. Find the shortest distance from $(2,0,2,0)$ to $U$.

This problem is in my textbook, and seem very easy, but I cannot imagine how can we find the distance from the point to the $span U$. Because $span U$ is not like a plane, a line... in geometry (in my thinking).

Thanks :)

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up vote 6 down vote accepted

The elements of U are in the form $(a,b,-b,-a)$. The distance from $(a,b,-b,-a)$ to $(2,0,2,0)$ is $(a-2)^2+b^2+(-b-2)^2+a^2$ Now minimize the last expression by minimizing $(a-2)^2+a^2$ and minimizing $(-b-2)^2+b^2$

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