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Homework Help: Why is this linearly independent?

  1. Apr 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Determine whether the given vector functions are linearly dependent or independent on the interval [tex](-\infty, \infty)[/tex]

    [tex]\begin{pmatrix} t \\ 3 \end{pmatrix}, \begin{pmatrix} 4 \\ 1 \end{pmatrix}[/tex]

    2. Relevant equations

    3. The attempt at a solution

    I computed the wronskian to be t-12. Since the wronskian is 0 at t=12, shouldn't this be linearly dependent? The book says that this is linearly independent. Why is that?
  2. jcsd
  3. Apr 18, 2008 #2
    The functions are only linearly dependent if the Wronskian is zero on the whole interval
  4. Apr 18, 2008 #3


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    Science Advisor

    Or, rather than using "high powered" stuff like the Wronskian, use the fact that two vectors are "dependent" if and only if one is a multiple of the other.
  5. Apr 18, 2008 #4
    This is basically the same as what HallsofIvy just suggested: Perform Gaussian Elimination on the matrix formed by combining the two vectors. Use the fact that the pivot columns of a matrix are linearly independent.
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