Matrix representation of $T$ with respect to $\{x,T(x),\dots,T^{n-1}(x)\}$

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.

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

1 Answer

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.

• So, you're saying that we can't write a matrix representation for this basis ? – Arman Malekzadeh Dec 9 '16 at 8:28
• 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. – Valentin Dec 9 '16 at 8:31
• I saw your other question. There you are given that $T^n(x)=0$, so you can write down all of the matrix representation. – Valentin Dec 9 '16 at 8:34
• @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. – amd Dec 9 '16 at 9:20