This definition of linear independence of functions is very confusing to me because what if functions fit neither definition over an interval.
Like number 5 says it's linearly dependent, but I feel like that's only true for specific numbers of t, not for the whole interval.
The functions from number 5 are linearly dependent because$$\color{red}{1}\times(t+1)+\color{red}{(-1)}\times(t^2+1)+\color{red}{1}\times(t^2-t)=0.$$Note the the numbers in red don't depend on $t$.