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Let $v_1, \dots, v_{n-1}$ be linearly independent vectors in $\mathbb{R}^n$. Their span defines a hyperplane; let $u$ be the unit normal vector to this hyperplane.

Now suppose we change $v_{11}$ (the first entry of $v_1$) to $v_{11} + \Delta$. How does $u$ change?

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What's your current line of thinking on this problem so far? –  Muphrid Nov 18 '12 at 5:43
    
It would be enough to come up with some sort of expression for $u$ in terms of the $v_j$, so that's my current plan of attack. I imagine the best way to do that is to solve the system of equations $v_j \cdot u = 0$ and $\|u\| = 1$, but I'm still bogged down on the details of that. –  GMB Nov 18 '12 at 5:58
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Form an $n \times (n-1)$ matrix with columns $v_1, \ldots, v_{n-1}$. Write it as $\pmatrix{w\cr W\cr}$ where $w$ is the first row and $W$ is $(n-1) \times (n-1)$. Correspondingly let $u^T = (a, b)$. Then we need $a w + b W = 0$. Assuming $W$ is invertible, we have $b = - a w W^{-1}$, i.e. $u^T = a (1, -w W^{-1})$. Since we want $u$ to be a unit vector, we take $|a| = \sqrt{1 + \|w W^{-1}\|^2}$. Adding $\Delta$ to $v_{11} = w_1$ subtracts the first row of $W^{-1}$ from $-w W^{-1}$; we then have to adjust $a$ to maintain $u$ as a unit vector.

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