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BenBlackburn

  • 2 years ago

how do you form a particular solution in linear algebra when you have more unknowns then equations?

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  1. TuringTest
    • 2 years ago
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    you mean a single solution? not possible... can you give an example?

  2. BenBlackburn
    • 2 years ago
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    yep so i have three equation w+2x+5y-5z =1 w+3x+10y-3z =2 x+4y+16z=-1 and im told to find the general solution. I know that this is the particular and the homogenious solution but i am unsure how to get them in this case

  3. TuringTest
    • 2 years ago
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    oh I think you are mixing up terminology go gauss-jordan elimination first and see what you can reduce this too

  4. BenBlackburn
    • 2 years ago
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    ok i did that and i got

  5. BenBlackburn
    • 2 years ago
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    w+2x+5y-5z=1 1x+5y+2z=1 -1y+14z=-2

  6. TuringTest
    • 2 years ago
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    reduce it all the way

  7. BenBlackburn
    • 2 years ago
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    see that is where i get stuck

  8. TuringTest
    • 2 years ago
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    you have to keep going do you know how to do this from matrix form? that is the most efficient....

  9. BenBlackburn
    • 2 years ago
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    ya i do i just dident know the best way for me to type it. do you mean reduced row echelon form?

  10. TuringTest
    • 2 years ago
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    yes

  11. TuringTest
    • 2 years ago
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    and it's true, you can't type it unless you know latex

  12. TuringTest
    • 2 years ago
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    the equation editor is too limited

  13. TuringTest
    • 2 years ago
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    \[\begin{array}9w+2x+5y-5z =1\\w+3x+10y-3z =2\implies\\~~~~~~~~~x+4y+16z=-1\end{array}\left[\begin{matrix}1&2&5&-5&|1\\1&3&10&-3&|2\\0&1&4&16&|-1\end{matrix}\right]\]

  14. TuringTest
    • 2 years ago
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    \[\left[\begin{matrix}1&2&5&-5&|1\\1&3&10&-3&|2\\0&1&4&16&|-1\end{matrix}\right]\implies\left[\begin{matrix}1&2&5&-5&|1\\0&1&5&2&|1\\0&1&4&16&|-1\end{matrix}\right]\]\[\left[\begin{matrix}1&0&-5&-11&|-1\\0&1&5&2&|1\\0&0&-1&14&|-2\end{matrix}\right]\implies\left[\begin{matrix}1&0&0&-81&|9\\0&1&0&72&|-9\\0&0&-1&14&|-2\end{matrix}\right]\]I hope I did that right

  15. BenBlackburn
    • 2 years ago
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    alright so how do you get from reduced row echelon to a general solution?

  16. TuringTest
    • 2 years ago
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    so we got all the way to\[\left[\begin{matrix}1&2&5&-5&|1\\1&3&10&-3&|2\\0&1&4&16&|-1\end{matrix}\right]\implies\left[\begin{matrix}1&2&5&-5&|1\\0&1&5&2&|1\\0&1&4&16&|-1\end{matrix}\right]\]\[\left[\begin{matrix}1&0&-5&-11&|-1\\0&1&5&2&|1\\0&0&-1&14&|-2\end{matrix}\right]\implies\left[\begin{matrix}1&0&0&-81&|9\\0&1&0&72&|-9\\0&0&1&-14&|2\end{matrix}\right]\](I fixed the minus sign so now this is complete) translate this back into letters and we have\[\begin{array}0w-81z=9\\x+72z=-9\\y-14z=2\end{array}\]so everything is dependent on \(z\) (note we could have done this so that it was a different variable, but we will get the same solution space) so what we do is we say that this represents a solution space depending on the value of the parameter \(z\) so we rename \(z\) as \(t\) and say that \[\begin{array}0w=81t+9\\x=-72t-9\\y=14t+2\end{array}\text{ where }t\in\mathbb R\]and that is our solution space in \(\mathbb R\)

  17. TuringTest
    • 2 years ago
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    I seem to have made a mistake somewhere in the matrix, but the concept is right here is the solution as wolfram has it http://www.wolframalpha.com/input/?i=solve+w%2B2x%2B5y-5z++%3D1+%2Cw%2B3x%2B10y-3z+%3D2%2Cx%2B4y%2B16z%3D-1+for+x%2Cy%2Cw though they do not rename z that is common practice to stress the dependence of each variable on a single real parameter not also that we could have solved for the other 3 variables as well instead

  18. BenBlackburn
    • 2 years ago
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    ah ok i see Thank you

  19. TuringTest
    • 2 years ago
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    welcome :)

  20. datanewb
    • 2 years ago
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    Ben, I think you are following the lectures from Strang's OCW. I wanted to point out that to get a particular solution, one easy way is to set the free variables to zero. This could be done as soon as the matrix A has been reduced to U... w+2x+ 5y- 5z= 1 1x+ 5y+ 2z= 1 -1y+14z=-2 or in terms of matrices:\[\left[\begin{matrix}1&2&5&-5\\ 0 &1&5&2\\0&0&-1&14\end{matrix}\right]\left[\begin{matrix}w\\x\\y\\z\end{matrix}\right]=\left[\begin{matrix}1\\1\\-2\end{matrix}\right]\] Setting the free variable (in this case z), you can use back substitution to calculate z = 0, y = 2, x = -9, w = 9... this is one particular solution. Strang calls it x-particular. Then adding all possible solutions to Ax = 0, N(A) gives all of the solutions to Ax = b. If you continue to reduce U to R, then the augmented matrix A|b looks like: \[\left[\begin{matrix}1&0&0&-79&|9\\0&1&0&72&|\neg9\\0&0&1&\neg14&|2\end{matrix}\right]\] And, you can simply read the solutions to x-particular off of b. Finally, setting the free variable, z, to 1, the N(A) is also easily discerned from R. z = 1, y = 14, x=-72, w=79. So the full solution would look something like this in vector form: \[\left[\begin{matrix}0\\2\\-9\\9\end{matrix}\right] + c\left[\begin{matrix}79\\-72\\14\\1\end{matrix}\right]\]

  21. datanewb
    • 2 years ago
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    This answer is equivalent to the one provided by TuringTest, but presented differently: the way Strang presented it.

  22. TuringTest
    • 2 years ago
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    sorry, I was wondering what they meant by "particular solution" in this case it clearly is asking for what @datanewb stated. thanks for finishing this problem correctly :)

  23. datanewb
    • 2 years ago
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    No problem. Thank you. Also, following your explanation helped round out my understanding.

  24. BenBlackburn
    • 2 years ago
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    Should the particular be [2,-9,9,0]?

  25. colorful
    • 2 years ago
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    just take the particular case of \(c=0\) and you get \[\left[\begin{array}00\\2\\-2\\9\end{array}\right]\]

  26. datanewb
    • 2 years ago
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    colorful is right. \[\left[\begin{matrix}2\\-9\\9\\0\end{matrix}\right]\] is just one of infinitely many particular solutions. It is not more or less correct than if we had chosen \[\left[\begin{matrix}79\\-70\\5\\10\end{matrix}\right]\] as the particular solution. Choosing the particular solution in which all of the free variables are equal to zero, is the most efficient way to find a particular solution.

  27. BenBlackburn
    • 2 years ago
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    ah alright thank you

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