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Go Classes | DA Course | Machine Learning

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Which of the following is a least square solution $\hat{\mathrm{x}}$ to the equation

 

$$ 
\left[\begin{array}{cc}
1 & -2 \\
2 & 1 \\
1 & -2 \\
2 & 1
\end{array}\right] \mathbf{x}=\left[\begin{array}{l}
1 \\
3 \\
1 \\
3
\end{array}\right] 
$$

  1.  $\left[\begin{array}{c}11 / 9 \\ 1 / 9\end{array}\right]$ 
  2.  $\left[\begin{array}{l}3 / 2 \\ 1 / 2\end{array}\right]$
  3.  $\left[\begin{array}{c}7 / 5 \\ 1 / 5\end{array}\right]$
  4.  $\left[\begin{array}{l}2 \\ 1\end{array}\right]$

2 Answers

0 0 votes
$\hat{\beta} = (X^TX)^{-1}X^TY$

$X^TX = \left[\begin{array}{cc} 10 & 0 \\  0 & 10 \end{array}\right]$

$(X^TX)^{-1} = \left[\begin{array}{cc} \frac{1}{10} & 0 \\  0 & \frac{1}{10} \end{array}\right]$

$X^TY = \left[\begin{array}{cc} \frac{1}{10} & \frac{2}{10} & \frac{1}{10} & \frac{2}{10} \\  \frac{-2}{10} & \frac{1}{10} & \frac{-02}{10} & \frac{1}{10} \end{array}\right]$

$\hat{\beta} =  \left[\begin{array}{cc} \frac{7}{5} \\ \frac{1}{5} \end{array}\right]$
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