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Determining Which Variables Are Free

  1. Feb 22, 2014 #1
    1. The problem statement, all variables and given/known data
    Find the column of the matrix in Exercise 39 that can be deleted and yet have the remaining matrix columns span [itex]\mathbb{R}^4[/itex]


    2. Relevant equations
    The matrix from problem 39 is

    [itex]\begin{bmatrix}
    10 & -7 & 1 & 4 & 6 \\
    -8 & 4 & -6 & -10 & -3 \\
    -7 & 11 & -5 & -1 & -8 \\
    3 & -1 & 10 & 12 & 12 \\
    \end{bmatrix}

    [/itex]



    3. The attempt at a solution

    I had matlab compute the row-reduced echelon form of the matrix augmented with the arbitrary point/vector [itex]\mathbf{b}[/itex], where [itex]\mathbf{b} \in \mathbb{R}^4[/itex]. The result is given as an attachment.

    I was wondering, does this augmented matrix contain any free variables. I believe it does, and allow me to explain:

    The augmented matrix can be written as

    [itex]\begin{bmatrix}
    1 & 0 & 0 & 1 & 0 & f_1(b) \\
    0 & 1 & 0 & 1 & 0 & f_2(b) \\
    0 & 0 & 1 & 1 & 0 & f_3(b) \\
    0 & 0 & 0 & 0 & 1 & f_4(b) \\
    \end{bmatrix}[/itex]

    Writing out the system of linear equations, that this augmented matrix represents, gives

    [itex]\begin{array} \\
    x_1 & + & x_2 & + & x_3 & + & x_4 & = & f_1(b) \\
    & & x_2 & + & x_3 & & & = & f_2(b) \\
    & & & & x_3 & + & x_4 & = & f_3(b) \\
    & & & & & & x_4 & = & f_4(b) \\

    \end{array}[/itex]

    If I understand what a free variable is, it is one of which can be written in terms of the others. In this example, we have the other variables can be written as a function of the variables x_4. So, we the variable x_4 is free because you can choose any value of it, and find the corresponding b_i's will be; then from these b_i's, x_3, x_2, and x_1 can be determined.

    Am I wrong?
     

    Attached Files:

    Last edited: Feb 22, 2014
  2. jcsd
  3. Feb 22, 2014 #2

    Ray Vickson

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    Homework Helper

    I your augmented form is right then your answer is also right.
     
  4. Feb 22, 2014 #3
    So, then x_4 is the only free variable?
     
  5. Feb 22, 2014 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    It is the free variable for the particular row and column ordering you have selected. If you permute the columns you might come up with a different free variable.

    I don't think looking at free variables is the way to go in this question. You want to know which four columns of the matrix form a basis, so the question you need to answer is this: if you omit one of the columns, do you have an invertible matrix? Your final echelon form has the identity matrix in columns 1,2,3,5, so if you leave out column 4 you get the identity matrix. That means that columns 1,2,3,5 form a non-singular matrix.
     
  6. Feb 22, 2014 #5
    No, I realize that determining which variables are free won't aid in my solving the actual problem. I was simply curious. Thank you.
     
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