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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:

  1. If $n = 4$, calculate the matrix of $T$ w.r.t the basis.
  2. 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!

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Starting with your last question, yes, "$M$ is the matrix of $T$" means "for all $u$, $T(u)=Mu$." –  Gerry Myerson Dec 14 '11 at 3:30
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1 Answer

up vote 2 down vote accepted

Yes, $T^2(v)$ means $T(T(v))$ in general $T^k(v)$ is composition of $T$ with itself $k$ times. And you are correct about how we should go about this problem, compute how the transformation acts on the basis. I will describe the process for how to do the first part below, and to do the second part you will do the exact same thing, but replace the transformation $T$ with the transformations $T^n$ for each $n$.

For $n = 4$ we have the basis as $$ v_1 = T^3(v), v_2 = T^2(v), v_3 = T(v), v_4=v$$

To compute the matrix of a transformation with respect to an ordered basis we simply compute how the transformation acts on that basis. In this case we have $$T(v_1) = T(T^3(v)) = T^4(v) = 0$$ by our assumption that $T^n(v) = 0$ for each $n$. Then we have $$T(v_2) = T(T^2(V)) = T^3(V) = v_1,$$ $$T(v_3) = T(T(v)) = T^2(v) = v_2$$ and $$T(v_4) = T(v) = v_3$$

So now we compose these into a matrix and get

$$[T]_{(v_1, v_2, v_3, v_4)} = \left(\begin{smallmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{smallmatrix}\right)$$

You should be able to preform the same process for the transformation $T^n$ for each $n$ required in part (b).

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Makes total sense, thanks a lot! –  John Doe Dec 14 '11 at 4:14
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