Linear Algebra Help: Change of Basis Matrix

Question

Let $T: V \rightarrow V$ be a linear transformation of the $n$-dimensional vector space $V$, and suppose there is a vector $v \in V$ such that the set $B = (v, Tv, T^2v,\ldots,T^{n-1}v)$ is a basis of $V$. let $f$ be the characteristic polynomial of $T$

Find the $B$ matrix $[T]_B$ of $T$ in terms of the $B$-coordinates of $T^nv$.

Answer 1

To get each column of $[T]_B$, you find the values of $T$ on the basis and rewrite them as column vectors in the given basis. Each column vector forms a column of your matrix. For example, the first column of the matrix is $Tv$, but written in the basis given. As a row vector this amounts to $(0,1,0,\ldots,0)$, because $$ Tv = 0\cdot v + 1\cdot Tv + 0 \cdot T^2v + \cdots + 0\cdot T^{n-1}v. $$

Answer 2

The answer is a companion matrix whose last column contains the $B$-coordinates of $T^n v$.