Check proof that two functions are linearly independent. I am not very experienced with math and need help with a proof. I need to show that the pair of functions $t$ and $1/t$ defined for $t&gt;0$ are linearly independent. Is the following a sufficient proof? If so, is there a simpler way?
Assume $at+b\frac{1}{t}=0$ is correct for any $t&gt;0$. Substituting $t=1$ yields $a+b=0$ (Eq. 1). Substituting $t=2$ yields $2a+\frac{b}{2}=0$ (Eq. 2). Subtracting Eq. 1 from Eq. 2 yields $a-\frac{b}{2}=0$, so $a=\frac{b}{2}$. Substituting $a=\frac{b}{2}$ into Eq. 1 yields $\frac{3}{2}b=0$, therefore $b=0$, which implies $a=0$.
Since the only scalars $a, b$ such that $at+b\frac{1}{t}=0$ are $a=b=0$ then $t$ and $1/t$ are linearly independent.
Your proof is fine and straightforward.
Maybe little shorter proof would be to see that $\lim_{t\rightarrow\infty} at+b\frac{1}t$ can be zero if and only if $a=0$. It follows immediately that $b=0$ too.
Well they are actually not linear independent. Take $t=1$ and $a=1;b=-1$
and You have $1+(-1*1/1$)
@billiam Thanks for your answer, if it is only positive, then it is quite trivial, here is my previous answer. Well, I don't know whether it may help, but since $t&gt;0;1/t&gt;0$ therefore $t+1/t$ can only be zero if the coefficients are zero.
But i could prove that $1/t$ is at least poaitive. first lets show that if $0&lt;x |*x \Leftrightarrow0&lt;x^2$ so the sqaure is always positive. Now $0&lt;t|*(1/t)^2\Leftrightarrow0* (1/t)^2&lt;t*(1/t)^2 \Leftrightarrow0&lt;1/t$ hope that helps
For any positive scalars the functions as it takes only positive values is positive again.