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Math Help - Linear algebra

  1. #1
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    Linear algebra

    Suppose that you are given the following system of linear equations
    [variables are x1, x2, x3 and x4]

    x1 + 3x2 + 9x3 + 2x4 =5
    x1 + 3x3 - 4x4 =5
    x2 + 2x3 + 3x4 =-1
    -2x1 + 3x2 + 5x4 =-1

    What is the general solution for the system? Write this general solution in
    parametric vector form.

    i row reduced the augmented matrix into:

    1 0 3 0 1
    0 1 2 0 2
    0 0 0 0 -1
    0 0 0 0 0

    did i row reduce this matrix correctly?

    the solution i got is
    x1=1-3x3
    x2=2-2x3
    x3 is free
    x4=-1
    any mistakes?

    I don't really know the difference between general solution and general solution in parametric vector form, can anybody show me the two? thanks.
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  2. #2
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    Quote Originally Posted by Vidal View Post
    the solution i got is
    x1=1-3x3
    x2=2-2x3
    x3 is free
    x4=-1
    any mistakes?
    .
    Assuming that you got the row reduced matrix right then your parametrized solutions are correct.

    Now let x3 = t where t is any real number to get,

    x1 = 1 - 3t
    x2 = 2 - 2t
    x3 = t
    x4 = -1
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Vidal View Post
    Suppose that you are given the following system of linear equations
    [variables are x1, x2, x3 and x4]

    x1 + 3x2 + 9x3 + 2x4 =5
    x1 + 3x3 - 4x4 =5
    x2 + 2x3 + 3x4 =-1
    -2x1 + 3x2 + 5x4 =-1

    What is the general solution for the system? Write this general solution in
    parametric vector form.

    i row reduced the augmented matrix into:

    1 0 3 0 1
    0 1 2 0 2
    0 0 0 0 -1
    0 0 0 0 0

    did i row reduce this matrix correctly?

    the solution i got is
    x1=1-3x3
    x2=2-2x3
    x3 is free
    x4=-1
    any mistakes?

    I don't really know the difference between general solution and general solution in parametric vector form, can anybody show me the two? thanks.
    TPH gave the general solution, here's the parametric form:
    Attached Thumbnails Attached Thumbnails Linear algebra-paraform.gif  
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  4. #4
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    Quote Originally Posted by Vidal View Post
    Suppose that you are given the following system of linear equations
    [variables are x1, x2, x3 and x4]

    x1 + 3x2 + 9x3 + 2x4 =5
    x1 + 3x3 - 4x4 =5
    x2 + 2x3 + 3x4 =-1
    -2x1 + 3x2 + 5x4 =-1

    What is the general solution for the system? Write this general solution in
    parametric vector form.

    i row reduced the augmented matrix into:

    1 0 3 0 1
    0 1 2 0 2
    0 0 0 0 -1
    0 0 0 0 0

    did i row reduce this matrix correctly?

    the solution i got is
    x1=1-3x3
    x2=2-2x3
    x3 is free
    x4=-1
    any mistakes?

    I don't really know the difference between general solution and general solution in parametric vector form, can anybody show me the two? thanks.
    There is a typo in the reduced augmented matrix in the third row. It should be

    1 0 3 0 1
    0 1 2 0 2
    0 0 0 1 -1
    0 0 0 0 0

    If the third row were

    0 0 0 0 -1

    as you have it, it would say 0 = 1 and the equations would be inconsistent with no solutions. Everything else is OK.
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  5. #5
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    thanks everybody. There is another part to this question which i don't know how to do.

    Let m be the number of linearly independent columns of A, let k be the number of parameters (free variables), and let n be the total number of columns in A. In our example above, n = 4.

    Notice that m + k = n. Do you suppose that this relationship will be true for all systems of linear equations? Why or why not?
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