Matrix representing a linear transformation

#linear_algebra

Let $V$ be a vector space with basis $v_{0}, \dots, v_{n}$ and let $a_{0}, \dots, a_{n}$ be scalars. Define a linear transformation $T : V \to V$ by the rules $T (v_{i}) = v_{i + 1}$ if $i < n$ , and $T (v_{n}) = a_{0} v_{0} + a_{1} v_{1} + \dots + a_{n} v_{n}$ . You don't have to prove this defines a linear transformation. Determine the matrix for $T$ with respect to the basis $v_{0}, \dots, v_{n}$ , and determine the characteristic polynomial of $T$ .

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Let $V$ be a vector space with basis ${\bf v}_0,\ldots, {\bf v}_n$ and let $a_0,\ldots, a_n$ be scalars. Define a linear transformation $T:V\to V$ by the rules $T({\bf v}_i)={\bf v}_{i+1}$ if $i<n$, and $T({\bf v}_n)=a_0{\bf v}_0+a_1{\bf v}_1+\cdots +a_n {\bf v}_n$. You don't have to prove this defines a linear transformation. Determine the matrix for $T$ with respect to the basis ${\bf v}_0,\ldots, {\bf v}_n$, and determine the characteristic polynomial of $T$.