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Why is this linearly independent?

  • Thread starter amolv06
  • Start date
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1. Homework Statement

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. Homework 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?
 

Answers and Replies

454
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The functions are only linearly dependent if the Wronskian is zero on the whole interval
 
HallsofIvy
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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.
 
19
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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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