Is Wolfram Alpha linear independence wrong or am I missing something?

Question

Maybe it's because you can't ask those questions to wolfram & I should use a matrix instead but when imputting

linear independence {$t$, $t^2+1$, $t^2+1-t$}

It says the three functions are linearly independent when the third one is clearly a linear combination of the other two. How should I input this to get a valid answer? I wanna check whether or not my results are correct.

Answer 1

It's because WolframAlpha interprets your input as one vector, i.e. the space of the single vector $(t, t^2+1, t^2+1-t)$.

An appropriate input would be (treating $1$, $t$ and $t^2$ as basis vectors):

linear independence (0,1,0), (1,0,1), (1,-1,1)

which outputs linearly dependent.

You can find other input examples for linear algebra here.

Answer 2

Another way to check for linear independence in W|A is to compute the Wronskian, say with the input "wronskian(($t$, $t^2+1$, $t^2+1-t$), $t$)", which results in $0$ so the set of functions is indeed linearly dependent.