Assume that $V$ is a vector space of dimension $n$ and $T:V \to V$ is a linear transformation.
Assume that $\{x,T(x),\dots,T^{n-1}(x)\}$ is a basis of $V$.

How can we write the matrix representation of $T$ with respect to this basis?

Note 1 : The method we learned in class for calculating matrix representation of a vector space with respect to a basis, was that we wrote $T(A)$ for all $A$ in the basis. But here, when i try, i don't know how to show $T(T(x))$ like a linear combination of members of $\{x,T(x),\dots,T^{n-1}(x)\}$. Is $T(T(x))=T^2(x)$ true? Why?

Note 2 : Yesterday, i asked a similar question. But notice that this question is just similar, not the same. So, its not a duplicate.

  • $\begingroup$ Well, what is $TB$, where $B$ is the matrix with these vectors as columns? $\endgroup$ – amd Dec 9 '16 at 8:18
  • $\begingroup$ @amd how do you multiply a linear transformation ( a function ) into a matrix?! $\endgroup$ – Arman Malekzadeh Dec 9 '16 at 8:19
  • $\begingroup$ Replace each column with its image, just like you describe in note 1. $\endgroup$ – amd Dec 9 '16 at 8:20
  • $\begingroup$ @amd So, you're saying that $TB=T(B)$? $\endgroup$ – Arman Malekzadeh Dec 9 '16 at 8:21

Note that $x \mapsto T(x)$ , $T(x) \mapsto T^2(x),T^2(x) \mapsto T^3(x), ...$

So each basis vector gets mapped to the next one. Try to visualize that in your matrix. We only don't know how to represent the image $T^n(x)$ of the last basis vector $T^{n-1}(x)$. So you can explicitely write down your matrix except for the last column.

  • $\begingroup$ So, you're saying that we can't write a matrix representation for this basis ? $\endgroup$ – Arman Malekzadeh Dec 9 '16 at 8:28
  • $\begingroup$ In general we can not know how to express $T^n(x)$ in terms of $\{x,T(x),T^2(x),...,T^{n-1}(x)\}$, so we can not write down the last column of the matrix wrt this basis. $\endgroup$ – Valentin Dec 9 '16 at 8:31
  • $\begingroup$ I saw your other question. There you are given that $T^n(x)=0$, so you can write down all of the matrix representation. $\endgroup$ – Valentin Dec 9 '16 at 8:34
  • $\begingroup$ @ArmanMalekzade As this appears to be a follow-on to your previous question, is $T^n(x)=0$ here, too? If so, then you should state that in the question. If not, then as Valentin points out, without knowing the value of $T^n(x)$ this question can’t be answered fully. $\endgroup$ – amd Dec 9 '16 at 9:20

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