# Homework Help: Why is this linearly independent?

1. Apr 18, 2008

### amolv06

1. The problem statement, all variables and given/known data

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

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

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. Apr 18, 2008

### kamerling

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

3. Apr 18, 2008

### 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.

4. Apr 18, 2008

### 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.