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Find the point on the hyperplane $x^Tc = β $that is closest to the origin by Lagrange multiplier method.

What is hyperplane and how we obtain its origin ! I need a serious hint !!

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  • $\begingroup$ A hyperplane is like a plane in 3D Euclidean space, but in higher dimensions, so is one dimension smaller than the space itself, and can be identified by the direction orthogonal to it and a point within it. $\endgroup$ – Henry Dec 1 '15 at 7:37
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HINT: You need solve this problem $$ \min_{x\in D}\frac{1}{2}\|x\|^2,\qquad D=\{x\in\mathbb{R}^n~:~x^{T}c-\beta=0\}. $$ For this, define the Lagrangian by $$ L(x,\lambda)=\frac{1}{2}\|x\|^2-\lambda(x^{T}c-\beta). $$ Imposing the condition $L_{x}(x,\lambda)=0$, we have $x=\lambda c$. As $x$ must belong to $D$ we have that $\lambda=-\beta/\|c\|^2$. So $x=-\frac{\beta c}{\|c\|^2}$. Remark: Check all the details that I did not do.

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