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Y=mx+c to matrix vector?

  1. Aug 24, 2012 #1
    So I have these two straight line equations:

    [itex]y = -x + 1, y = 2x + 1[/itex] So now I have a single point of intersection of [itex](1y,0x)[/itex]

    But the next part of the question is what confuses me but I've included a picture as it's just easier... would someone please be able to take the time to explain each step in the question?

    I can do parts B and C but A is confusing me...
     

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  2. jcsd
  3. Aug 24, 2012 #2
    This would probably get a better response in the homework question forums.
     
  4. Aug 24, 2012 #3

    berkeman

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    Thread moved.
     
  5. Aug 24, 2012 #4

    LCKurtz

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    Write your two equations like this:
    ##x+y=1##
    ##-2x+y=1##
    Then put it in the form:$$
    \left[ \begin{array}{cc}
    ? & ?\\ ? & ?
    \end{array}\right]
    \left[ \begin{array}{c}
    x\\y
    \end{array}\right] = \left[ \begin{array}{c}
    ?\\?
    \end{array}\right]$$
     
    Last edited: Aug 24, 2012
  6. Aug 25, 2012 #5

    [itex]\left[ \begin{array}{cc}
    1 & 1\\ -2 & 1
    \end{array}\right]
    \left[ \begin{array}{c}
    x\\y
    \end{array}\right] = \left[ \begin{array}{c}
    1\\1
    \end{array}\right][/itex]

    That doesn't even look remotely correct :/
     
  7. Aug 25, 2012 #6

    HallsofIvy

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    Why do you say that? We had kind of assumed that, since you talked about matrices and vectors, you knew what they are and knew how to multiply a matrix and a vector. is that not true?

    Can you see that the matrix multiplication gives
    [tex]\begin{bmatrix}1 & 1 \\-2 & 1\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}x+ y\\ -2x+ y\end{bmatrix}= \begin{bmatrix}1 \\ 1\end{bmatrix}[/tex]
    and that two vectors are equal if and only if their corresponding components are equal.

    That is, that matrix-vector equation is exactly the same as saying that x+ y= 1 and -2x+ y= 1.
     
  8. Aug 25, 2012 #7

    I've multiplied matrices together and I know how to do them but usually I get given them in the form

    [tex]\begin{bmatrix}1 & 1 \\-2 & 1\end{bmatrix}\begin{bmatrix}2 & 4 \\ 3 & 1\end{bmatrix}[/tex] For example and then you just multiply the 2 matrices together. But I understand how you explained it thanks.
     
  9. Aug 25, 2012 #8
    OK I've got to the last stage, I've worked out the matrix inverse which is [tex]\begin{bmatrix}0.333 & -0.333 \\0.666 & 0.333\end{bmatrix}[/tex] and I've verified it using the Identity matrix [tex]\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}[/tex] so now I just have to plot the 2 straight lines but I'm kinda lost in my own work now...

    Am I plotting the original 2 straight lines using the equations or am I plotting them via the matrix and if so how do I do that?

    EDIT: It says using the matrix inverse but I'm not sure how to do it
     
  10. Aug 25, 2012 #9

    rock.freak667

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    You changed your equations into a matrix form of Ax=B

    so if you pre-multiply everything by A-1 you will have

    A-1Ax = A-1B

    And you worked out what A-1A was. So just work out the right side of the equation.
     
  11. Aug 25, 2012 #10

    LCKurtz

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    Get rid of the decimals. That is just an approximation to the inverse and it won't give you the identity when multiplying. There is absolutely no reason to use decimal approximations instead of exact fractional values in this problem.
     
  12. Sep 1, 2012 #11
    I'm still stuck on this question, I kinda just forgot about it but I really need all the marks I can get, can someone please tell me how to plot the point of intersection of the 2 lines using the inverse matrix?

    It is worth 5% so I need to know how to do it.
     
  13. Sep 1, 2012 #12

    LCKurtz

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    Did you read RockFreak's post #9?
     
  14. Sep 1, 2012 #13
    Yes but I don't understand what he means... I have worked out what A[itex]^{-}[/itex]1 is but I don't know how to plot the matrix elements as points on the graph
     
  15. Sep 1, 2012 #14

    rock.freak667

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    There is no need to plot a graph, you just need to calculate A-1B.
     
  16. Sep 1, 2012 #15

    LCKurtz

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    Why don't you just plot the two straight lines the old fashioned way and look at how the intersection point compares with what you get when you solve the matrix equation by multiplying both sides by A-1?
     
  17. Sep 1, 2012 #16
    What is B though?
     
  18. Sep 1, 2012 #17

    LCKurtz

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    Look at the matrices in your post #5. Use them.
     
  19. Sep 1, 2012 #18
    2, -1?
     
  20. Sep 1, 2012 #19

    LCKurtz

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    Do you really have to ask? Does that point work in both lines? If it does you are done, otherwise check your arithmetic.
     
  21. Sep 1, 2012 #20
    I'm am losing my temper with this crap now, I don't know what is right, if you don't know what you're doing then you don't know if the answer is right or not... this is not homework, I am revising for a test so please can you just tell me?
     
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