So I asked a question recently:
Finding the matrix of this linear transformation
and I'm wondering something else.
$T : V \to V$ is a linear transformation. There is a vector $v \in V$ such that $T^n(v) = 0$. We're also told that the vectors $T^{n-1}(v), T^{n-2}(v), \ldots, T(v), v$ form a basis for $V$.
Let's also add that $T^{n-1}(v) \neq 0$
What if we're not told what the basis is? How do you find a basis for the transformation, or at least prove that it is a basis?
In general, a set of vectors is a basis for a space if the vectors are linearly independent and they span the space. How can we show that if we're not given what the actual vectors are?
Edit: My question boils down to: How do I show that those vectors form a basis for V, if we don't even know what V is?