Under which condition are $(1,a)$ and $(1,b)$ linearly independent?
I am not completely sure if my logic is correct, but I know that linearly independent means that the vectors cannot be multiples of each other. So, I found that this is only true when $a \neq b$. Is this correct?
For only TWO vectors, you are "right": two vectors are linearly independent (L.I) if their coordinates are not propotional.
However, a more rigourous definition is
"Any set of vectors are linearly independent if the only linear combination of them that result in the void vector requires all coeficients to be 0".
More easily, you take the linear combination
$a_1\vec{v}_1+a_2\vec{v}_2+\dots = \vec{0}$
and now, vectors are L.I. if the only possibility is that all coefficients are 0 $(a_1=a_2=\dots=0$).
If that's not the case (if there are more possibilities with other numbers), the family of vectors is L.D.