We're given $V$, which is an $n$ dimensional vector space. $T : V \to V$ 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 $V$.
The questions are:
- If $n = 4$, calculate the matrix of $T$ w.r.t the basis.
- If $n = 4$, calculate the matrix for $T^n$ for $n = 2,3,4$.
From a previous question, I know that if you want to form the matrix for a transformation, you simply compute the value for $T$ at the basis, and express your answer as a matrix w.r.t the basis. But, we can't really "compute" in this case because we don't know what the actual transformation is.
Also, when they say $T^2(v)$, do they just mean $T(T(v))$? If so, I suppose $T(T^{n-1}(v)) = 0$, but I'm not sure what else we can figure out or even how to construct a matrix.
As for the second question, I don't really see what they're asking.
Lastly, say we do find a matrix, call it $M$. Let's say I have a vector $u$, and say $T(u) = s$, where $s$ is some other vector. Does the relationship $Mu = s$ always hold in this case?
Thanks a bunch for all your help!