Determine whether these functions are linearly independent by using the Wronskian.

Question

I have to solve the following exercise:

Use the Wronskian to check whether the given set of functions is linearly dependent:

$$f_1(t) = 2t -3, \,f_2(t) = 2t^2 +1, \,f_3(t) = 3t^2 + t$$

How should I solve this exercise? I only know how to use the Wronskian for a system that is at least $2\times 2$!

Edit: I understand that if $f_1$ and $f_2$ are of dimension $n$, $n>1$, that in order to check whether they're linearly dependent we can look at the determinant of the resulting system. What I don't understand is that if $f_1$ and $f_2$ are one-dimensional we can just look at the determinants of $f_1$ and $f_2$. I only need to show linear (in)dependence between $f_1$ and $f_2$, not between $f = [f_1, f'_1]$ and $g = [f_2,f'_2]$, right?

Answer 1

Form the matrix of derivatives,

$$\begin{bmatrix}f_1(t) & f_2(t) & f3(t)\\ f'_1(t) & f'_2(t) & f'_3(t)\\ f''_1(t) & f''_2(t) & f''_3(t)\end{bmatrix}$$

and then take the determinant. If the determinant of this matrix, which is what the wronskian actually is, is not identically zero on an interval then the functions are linearly independent on this interval.

Answer 2

It's just the same except it has three lines and three columns $$W=\left |\pmatrix{f_1 &f_2 & f_3 \\f_1' &f_2' & f_3' \\f_1'' &f_2'' & f_3'' }\right |=\left |\pmatrix{2t-3 &2t^2+1 & 3t^2+t\\2 &.. & ... \\0 & 4 & 6 }\right |$$

Note that:

$$f_3(t) = \frac 3 2 f_2+\frac 12f_1$$

So you already now the answer