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2 years ago

Just some questions I need answered!

Mathematics

1 answer:

Anni [7]2 years ago

7 0

4z>-60

divideby 4

z>-15

x+19 <=-5

subtract 19 from each side

x<= -24

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Formulate the following problem as least squares problems. For each problem, give a matrixA and a vector b such that the problem

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a) $A=\left[\begin{array}{ccc}1&2&3\\1&-1&1\end{array}\right]$

$b=\left[\begin{array}{ccc}0\\1\end{array}\right]$

b) $||Ax-b||^{2} =(-bx_{2}+4)^{2} (-4x_{1} +3x_{2} -1)^{2} +(x_{1} +8x_{2} -3)^{2}$

c) $A=\left[\begin{array}{ccc}0&6\sqrt{2} &0\\\sqrt{3} &3\sqrt{3} &0\\2&-16&0\end{array}\right]$

$x=\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right]$

$b=\left[\begin{array}{ccc}-\sqrt{2} \\\sqrt{3} \\6\end{array}\right]$

Step-by-step explanation:

a) considering the equation:

Minimize $x_{1}^{2} +2x_{2}x^{2} +3x_{3}^{2}+(x_{1} -x_{2} +x_{3} -1)^{2} +(-x_{1} -4x_{2} +2)^{2}$

$A=\left[\begin{array}{ccc}1&2&3\\1&-1&1\end{array}\right]$ (matrix A)

vector b

$b=\left[\begin{array}{ccc}0\\1\end{array}\right]$

b) If Pxn is matrix B and p-vector d, we have:

minimize $(-6x_{2}+4)^{2} +(-4x_{1} +3x_{2} -1)+(x_{1} +8x_{2} -3)^{2}$

$Ax=\left[\begin{array}{ccc}0&-6&0\\-4&3&0\\1&8&0\end{array}\right]$

$\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right]$

$b=\left[\begin{array}{ccc}-4\\1\\3\end{array}\right]$

$Ax-b=\left[\begin{array}{ccc}-bx_{2}+4 \\-4x_{1}+3x_{2}-1 \\x_{1}+8x_{2}-3 \end{array}\right] =1$

$||Ax-b||^{2} =(-bx_{2}+4)^{2} (-4x_{1} +3x_{2} -1)^{2} +(x_{1} +8x_{2} -3)^{2}$

c) minimize $2(-bx_{2}+4)^{2} +3(-4x_{1} +3x_{2} -1)^{2} +4(x_{1} -x_{2} -3)^{2} -(6\sqrt{2}x_{2} +4\sqrt{2} )^{2} +(-4\sqrt{3} x_{1} +3\sqrt{3}x_{2} -\sqrt{3})^{2} +(2x_{1} -16x_{2} -6)^{2}$

in matrix:

$A=\left[\begin{array}{ccc}0&6\sqrt{2} &0\\\sqrt{3} &3\sqrt{3} &0\\2&-16&0\end{array}\right]$

$x=\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right]$

$b=\left[\begin{array}{ccc}-\sqrt{2} \\\sqrt{3} \\6\end{array}\right]$

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2 years ago