determine linear independece of these two functions

Question

How to determine whether the following two functions are linearly independent?

$f(t)= \sqrt{t}$ and $g(t)=\frac1{t}$

I know that they are linearly independent using the Wronskian. How would you calculate without using the Wronskian?

Answer 1

Without using the Wronskian, I would note that:

• $f(1)=1$;
• $f(4)=2$;
• $g(1)=1$;
• $g(4)=\frac14$.

So, if $\alpha,\beta\in\mathbb R$ are such that $\alpha f+\beta g=0$, then $\alpha\times1+\beta\times2=\alpha\times2+\beta\times\frac14=0$. But the only solution of the system$$\left\{\begin{array}{l}\alpha+2\beta=0\\2\alpha+\frac\beta4=0\end{array}\right.$$is $\alpha=\beta=0$.

Answer 2

If they were not, you would get $(a, b) \neq (0,0)$ such that $$a f(t) + b g(t) = 0 \quad \forall t > 0$$

Evaluate in $t=1$ and $t=4$ (for example), you get $a+b = 0 \quad \text{and} \quad 2a + \frac{b}{4} = 0$$

Solving the system leads you to $a=b=0$. Absurd.