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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

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2 Answers 2

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Think about what it means for T to be non-invertible. What does that tell you about its range as compared to its domain? You'll need to refer back to the definition of vector space and subspace as well and remember that there may be multiple valid bases of a given space.

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  • $\begingroup$ Ok I think I got part (a) using your hints. But for part b, do we have enough info to find the B of W and C of V? $\endgroup$
    – user34166
    Commented Nov 20, 2013 at 15:21
  • $\begingroup$ For my first post I only read (a), so if you only got (a) with that hint....that makes sense! $\endgroup$
    – rcorty
    Commented Nov 20, 2013 at 15:23
  • $\begingroup$ Sorry, I don't know how to answer (b). $\endgroup$
    – rcorty
    Commented Nov 20, 2013 at 15:25
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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?

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