Why is this linearly independent?

[amolv06]

Homework Statement

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

\begin{pmatrix} t \\ 3 \end{pmatrix}, \begin{pmatrix} 4 \\ 1 \end{pmatrix}

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?

 

[kamerling]

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

 

[HallsofIvy]

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.

 

[White Ink]

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.