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Math Help - Using matrices to solve systems of linear equations.

  1. #1
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    Using matrices to solve systems of linear equations.

    Hello,

    Could someone please tell me how to solve exercises 11, 22 and 26?

    Exercise 11
    The answer of exercise 11 should be that b is not a linear combination of a1, a2 and a3. I computed the echelon form of the augmented matrix where a1, a2, a3 and b are the columns of the augmented matrix, but I end up with a consistent system, so I don't understand why b is not a linear combination of a1, a2 and a3.

    Exercise 22
    Not even a clue where to start...???

    Exercise 26
    In 26(a) I found that the system is consistent, so b is in W. How do I solve 26(b)?

    Thanks for your help!

    Lotte
    Attached Thumbnails Attached Thumbnails Using matrices to solve systems of linear equations.-section1.3exercise11.png   Using matrices to solve systems of linear equations.-section1.3exercise22.png   Using matrices to solve systems of linear equations.-section1.3exercise26.png  
    Last edited by mr fantastic; November 14th 2011 at 11:04 AM. Reason: Re-titled.
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  2. #2
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    Re: Linear Algebra Help!

    when i reduce the augmented matrix in #11, i get:

    \begin{bmatrix}1&0&5&2\\0&1&4&3\\0&0&0&-15\end{bmatrix}

    the last row implies 0a_1 + 0a_2 + 0a_3 = 0 = -15

    doesn't look consistent to me.
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  3. #3
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    Re: Linear Algebra Help!

    Where did I go wrong?

    \begin{bmatrix}1&0&5&2\\-2&1&-6&-1\\0&2&8&6\end{bmatrix}

    (Add 2 times the 1st row to the 2nd row)

    \begin{bmatrix}1&0&5&2\\0&1&4&3\\0&2&8&6\end{bmatr  ix}

    (Add -2 times the 2nd row to the 3rd row)

    \begin{bmatrix}1&0&5&2\\0&1&4&3\\0&0&0&0\end{bmatr  ix}

    Could you please help me with the other exercises aswell?
    Last edited by Lotte1990; November 15th 2011 at 03:56 AM.
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  4. #4
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    Re: Linear Algebra Help!

    I used MATLAB to get the row reduced echelon form of this matrix. MATLAB gives the same row reduced echelon form I found.
    Does anyone know how Deveno ended up with an inconsistent row? The answer of exercise 11 should be that b is not a linear combination of a1, a2 and a3.
    I don't understand why, because the row reduced echelon form of the matrix shows a consistent system...

    Please help!
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  5. #5
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    Re: Linear Algebra Help!

    hmm...i must have made an error, because i re-checked and got the same matrix as you.

    so, let's see what that means:

    it means all vectors in col(A), are linear combinations of the first two columns of A (those are the pivot columns).

    suppose that (2,-1,6) = a(1,-2,0) + b(0,1,2) = (a,b-2a,b). then a = 2, and b = 6, but then b-2a = 6 - 4 - 2 ≠ -1.
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