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October 15th, 2018, 05:09 AM   #1
woo
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Linear system

Question: Using the fact that $\displaystyle \textbf{Ax=b}$ is consistent if and only if $\displaystyle \textbf{b}$ is a linear combination of the columns of $\displaystyle \textbf{A}$ to find a solution to

$\displaystyle \left( \begin{array}{cccc}
1 & 2 & 3 & 4 \\
2 & 3 & 4 & 1 \\
3 & 4 & 1 & 2 \end{array} \right)$$\displaystyle \left( \begin{array}{c}
x\\
y\\
z\\
w \end{array} \right)$$\displaystyle =\left( \begin{array}{c}
20\\
20\\
20\end{array} \right).$

My work: I rewrite the given system into

$\displaystyle x\left( \begin{array}{c}
1\\
2\\
3\end{array} \right)$$\displaystyle +y\left( \begin{array}{c}
2\\
3\\
4\end{array} \right)$$\displaystyle +z\left( \begin{array}{c}
3\\
4\\
1\end{array} \right)$$\displaystyle +w\left( \begin{array}{c}
4\\
1\\
2\end{array} \right)$$\displaystyle =\left( \begin{array}{c}
20\\
20\\
20\end{array} \right).$

What should I do next? Please give some hints. Thank you.
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October 15th, 2018, 08:51 AM   #2
SDK
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Math Focus: Dynamical systems, analytic function theory, numerics
Notice that all 3 of the rows sums to 10.
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October 15th, 2018, 09:15 AM   #3
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As, x = y = z = w = 2 is an obvious solution, the system is consistent.

You can use $\text{uAx = ub}$, where $\text{u}$ is the vector (1, -2, 1) if you want to find out more.
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October 16th, 2018, 08:35 AM   #4
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Quote:
Originally Posted by woo View Post
My work: I rewrite the given system into

$\displaystyle x\left( \begin{array}{c}
1\\
2\\
3\end{array} \right)$$\displaystyle +y\left( \begin{array}{c}
2\\
3\\
4\end{array} \right)$$\displaystyle +z\left( \begin{array}{c}
3\\
4\\
1\end{array} \right)$$\displaystyle +w\left( \begin{array}{c}
4\\
1\\
2\end{array} \right)$$\displaystyle =\left( \begin{array}{c}
20\\
20\\
20\end{array} \right).$

What should I do next? Please give some hints. Thank you.
x+2y+3z+4w=20
x+3y+4z+ w=20
3x+4y+z+2w=20

Put in augmented matrix form and sove by row reduction to get:

1 2 3 | 20 -4w
2 3 4 | 20 -w
3 4 1 | 20 -2w

1 2 3 | 20 -4w
0 1 2 | 20 -7w
0 0 -2| 0 -2w

z=w
y=20-11w
x=20-9w

$\displaystyle \begin{bmatrix}
x\\
y\\
z
\end{bmatrix}=\begin{bmatrix}
20-9w\\
20-11w\\
w
\end{bmatrix}=\begin{bmatrix}
20\\
20\\
0
\end{bmatrix}+w\begin{bmatrix}
-9\\
-11\\
1
\end{bmatrix}
$

You can ck the algebra.
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