# Linear transformation and characteristic polynomial

Let $V$ be an $n$-dimensional vector space and $T : V \to V$ a non-invertible linear transformation.

1. Show that there is a subspace $W \subset V$ which is $(n−1)$-dimensional and contains $\operatorname{range}T$.

2. Define $S : W \to W$ by $S(w) = T(w)$, viewed as a vector in $W$. Show that $p_T(\lambda) = −\lambda p_S(\lambda)$, where $p_S$ and $p_T$ are the characteristic polynomials of $S$ and $T$, respectively. (Hint: Extend a basis $B$ of $W$ to a basis $C$ of $V$. Compare $[T]C$ to $[S]B$.)

3. Show that if $p_T (\lambda) = (−\lambda)^n$ then $T^n = 0$. (Hint: Proceed by induction, using (a) and (b).)

I'm not sure where to start. Just a little guidance is required. Thanks

For part 2. do as the hint suggests: Extend a basis of $W$ to a basis of $V$. More precisely, you can chose the "extending vector" to be in $\operatorname{ker} T$. What does this say about the shape of $[T]_C$? Another hint: How do you compute the characteristic polynomial of a block diagonal matrix?