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$$\begin{array}{ll} A \mbox{\boldmath$x$} = \mbox{\boldmath$b... ... \mbox{\boldmath$b$} = {}^T\!(b_1,b_2,\ldots,b_m) \end{array}$$

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$$Err = \sum_{k=1}^{m}(\mbox{\boldmath$\alpha$}_k \mbox{\boldm... ...( \mbox{\boldmath$\alpha$}_k = (a_{k1},a_{k2},\ldots ,a_{kn}))$$

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$$\mbox{\boldmath$\beta$}_1,\mbox{\boldmath$\beta$}_2,\ldots,... ...mbox{\boldmath$\beta$}_k={}^T\!(a_{1k},a_{2k},\ldots,a_{mk}))$$

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$${}^T\!A A\mbox{\boldmath$x$} = {}^T\!A\mbox{\boldmath$b$}$$

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$$\left\{ \begin{array}{lll} ax+by & = & p\\ cx+dy & = & q\\ ex+fy & = & r \end{array} \right.$$

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$$A = \left[\begin{array}{ll}a & b\\ c & d \\ e & f\end{array... ...ath$x$}={}^T\!(x,y), \ \ \ \mbox{\boldmath$b$}={}^T\!(p,q,r)$$

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$$Err=(ax+by-p)^2+(cx+dy-q)^2+(ex+fy-r)^2$$

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$$\frac{\partial Err}{\partial x}= \frac{\partial Err}{\partial y}=0$$

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$$\begin{eqnarray*}\frac{\partial Err}{\partial x} & = & 2a(ax+by-p) + 2c(cx+dy-... ...rr}{\partial y} & = & 2b(ax+by-p) + 2d(cx+dy-q) + 2f(ex+fy-r) \end{eqnarray*}$$

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$$\begin{eqnarray*}&& \left[\begin{array}{l} a(ax+by-p) + c(cx+dy-q) + e(ex+fy-r... ...ath$b$}\\ & = & \left[\begin{array}{l}0\\ 0\end{array}\right] \end{eqnarray*}$$

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$$\begin{eqnarray*}\det({}^T\!A A) & = & \left\vert \left[\begin{array}{lll}a &... ...2+d^2+f^2)-(ab+cd+ef)^2\\ & = & (ad-cb)^2+(af-eb)^2+(cf-ed)^2 \end{eqnarray*}$$

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