There is a famous question from the book on Linear Algebra by Friedberg which says:
Let $V$ be a finite-dimensional vector space, and let $T: V → V$ be linear. Prove that $V = R(T^k) \oplus N(T^k)$ for some positive integer $k$.
My answer :
I first observed that for $n \in \mathbb{N}$, $$R(T^{n+1}) = T^{n+1}(V) = T^n(T(V)) = T^n(R(T)) \subseteq T^n(V) = R(T^n)$$ So, $R(T^{n+1}) \subseteq R(T^n) \ \ \forall \ n \in \mathbb{N}$ and hence $rank(T) \geq rank(T^2) \geq rank(T^3) \geq \cdots$.
So, as rank of a transformation must be a natural number, this implies that
$\exists \ \ k \in \mathbb{N}$ such that $\ rank(T^m) = rank(T^k) \ \ \forall \ \ m \geq k$
$\implies rank (T^k) = rank(T^{2k})$.
Now we know:
- for $U : X \to Y$, $rank (U) = rank(U^{2}) \implies X = R(U) \oplus N(U)$
- and $T^k : R(T^{k-1}) \to V$
Hence, we get : $R(T^{k-1}) = R(T^k) \oplus N(T^k)$
My doubt :
What am I missing here ? How am I supposed to get $V = R(T^k) \oplus N(T^k)$ ?
P.S. Doubt cleared :
See the discussion in the comment box.