Why is this linearly independent?

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Homework Help Overview

The discussion revolves around determining the linear dependence or independence of two vector functions over the interval (-∞, ∞). The original poster presents a calculation involving the Wronskian and expresses confusion regarding the result compared to the textbook answer.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to use the Wronskian to assess linear dependence, questioning the interpretation of its value at a specific point. Other participants suggest alternative methods, such as checking for scalar multiples or using Gaussian elimination.

Discussion Status

The discussion is active, with participants exploring different interpretations of linear dependence. Some guidance has been offered regarding alternative approaches to the problem, but no consensus has been reached on the correct interpretation of the Wronskian's implications.

Contextual Notes

There is an underlying assumption that the Wronskian's value at a single point may not fully determine linear dependence over the entire interval. The participants are navigating this nuance in their reasoning.

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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]

Homework Equations


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?
 
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The functions are only linearly dependent if the Wronskian is zero on the whole interval
 
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.
 
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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