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Math Help - Proof with System of Linear Equations

  1. #1
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    Proof with System of Linear Equations

    The curve y=ax^2 + bx + c passes through the points (x_1, y_1), (x_2, y_2), (x_3, y_3). Show that the coefficients a, b, and c are a solution of the system of linear equations whose augmented matrix is:

    \begin{bmatrix} x^2_1& x_1& 1& y_1 \\<br />
x^2_2& x_2& 1& y_2 \\<br />
x^2_3& x_3& 1& y_3 \end{bmatrix}<br />

    Just began linear algebra and having trouble understanding what this question is asking. I thought only constants/coefficients could go in the augmented matrix, but the x's and y's are variables. Confusing.
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  2. #2
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    Can you write out three equations that you know the curve satisfies? I would then compare those three equations with matrix multiplication, and see if the result pops out at you.

    What goes into a matrix can be all sorts of things, depending on the context.
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  3. #3
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    Technically, what's in that matrix are all constants.
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  4. #4
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    Quote Originally Posted by Ackbeet View Post
    Can you write out three equations that you know the curve satisfies? I would then compare those three equations with matrix multiplication, and see if the result pops out at you.

    What goes into a matrix can be all sorts of things, depending on the context.
    I think the three equations are:
    ax^2_1 + bx_1 + c = y_1
    ax^2_2 + bx_2 + c = y_2
    ax^2_3 + bx_3 + c = y_3
    But we have not covered matrix multiplication yet. That is the next section of the book. Is there anyway to do this without matrix multiplication?
    Last edited by seuzy13; August 9th 2010 at 02:25 PM. Reason: Formatting was off
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  5. #5
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    Well, if you haven't done matrix multiplication yet, you must have some method of constructing the augmented matrix. In fact, you can read off the coefficients of the augmented matrix from the three (incidentally correct) equations you wrote out. Does that make sense?
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  6. #6
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    Quote Originally Posted by Ackbeet View Post
    Well, if you haven't done matrix multiplication yet, you must have some method of constructing the augmented matrix. In fact, you can read off the coefficients of the augmented matrix from the three (incidentally correct) equations you wrote out. Does that make sense?
    Yes, I think it's making sense now. The most confusing thing was accepting that x_1, etc. were acting as the constants/coefficients.
    Thanks!
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  7. #7
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    You're very welcome. Have a good one!
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