Linear Algebra – Proving V = ker(?^n) ? image(?^n)

jordan-normal-formlinear algebra

Let $V$ be a $n$-dimensional complex vector space and $\phi:V\to V$ a linear mapping. Prove that $$V = \ker(\phi^n) \oplus \text{image}(\phi^n)$$

Here is my attempt:

Since $\phi^n$ is also a linear mapping of $V$ into $V$, we have that $$\dim V = \dim \ker(\phi^n) + \dim \text{image}(\phi^n).$$
We only need only to show that this sum is direct, in other words, that $$\ker(\phi^n) \cap \text{image}(\phi^n) = \{0\}.$$ since this would imply $$V = \ker(\phi^n) + \text{image}(\phi^n)$$

We let $v \in \ker(\phi^n) \cap \text{image}(\phi^n)$ be arbitrary and aim to show that $v=0$. $\ker(\phi^n)$ is the generalized eigenspace of $\phi$ for the eigenvalue $0$, so there is a $k \leq n$ such that $\phi^k(v) = 0$.

This is where I'm stuck. How do I proceed from here? Is there a different way to do this?

Best Answer

We consider the chains $$V\subset\phi(V)\subset\cdots\subset\phi^n(V)\subset\cdots$$ and $$\dim(V)\geq\dim(\phi(V))\geq\cdots\geq\dim(\phi^n(V))\geq\cdots$$

  • If $\dim(\phi^n(V))=1$, it is easy to prove.

  • If $\dim(\phi^n(V))\geq2$, there exist $k\leq n$ such that $\phi^k(V)=\phi^{k+1}(V)=\cdots$. Actually, $\phi:\phi^n(V)\rightarrow\phi^n(V)$ is an isomorphism.

    • For any $a\in V$, there exists $b$ such that $\phi^n(a)=\phi^{2n}(b)$. Then we can prove $a=c+\phi^n(b)$ and $\phi^n(c)=0$.
    • We assume $a\in\ker(\phi^n)\cap\text{image}(\phi^n)$. That is to say, $\phi^n(a)=0$ and $\phi^n(b)=a$. If $a\ne0$, we get $\phi^{2n}(b)=0$ and $\phi^n(b)\ne0$ which will lead $\dim(\phi^n(V))>\dim(\phi^{2n}(V))$. That is a contraction.

Look at an another question Show that $V = \mbox{ker}(f) \oplus \mbox{im}(f)$ for a linear map with $f \circ f = f$.

  • In your question, $\phi:\phi^n(V)\rightarrow\phi^n(V)$ is an isomorphism. Also $\phi^n:\phi^n(V)\rightarrow\phi^n(V)$ is an isomorphism.

  • In another question, $f=id: f(V)\rightarrow f(V)$ is an isomorphism.

  • Lemma Let $f:V\rightarrow V$ be a linear map with $f:f(V)\rightarrow f(V)$ being an isomorphism. Then we have $$V=\ker(f)\oplus\text{image}(f)$$

Related Question