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 🙂
Best Answer
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 $(-b-2)^2+b^2$